Probability and Random Variables A FINANCIAL TIMES COMPANY
|
|
- Jasper Cook
- 5 years ago
- Views:
Transcription
1 Probability Basics
2 Probability and Random Variables A FINANCIAL TIMES COMPANY 2
3 Probability Probability of union P[A [ B] =P[A]+P[B] P[A \ B] Conditional Probability A B P[A B] = Bayes Theorem P[A \ B] P[B] P[B A] = P[B \ A] P[A] P[A \ B] =P[B \ A] Joint probability P[A B] = P[A \ B] P[B] = P[B A]P[A] P[B] Independence B P[B] =05 P[A B] =P[A]! P[A \ B] =P[A]P[B] A P[A] =05 P[A \ B] =025 3
4 Probability Mutually exclusive events P[A [ B] =P[A]+P[B]! P[A \ B] =0 A B Mutually exclusive events are not independent! Collectively exhaustive events P[A [ B] =1 P[A]+P[B] 1 Mutually exclusive and collectively exhaustive events P[A [ B] =1 P[A]+P[B] =1 P[B] =P[Ā] A B 4
5 Discrete Random Variables Value of a discrete random variable, X, is determined by the outcome (event) of a random experiment that generates discrete outcomes, eg a coin toss Sample space of a random experiment is the set of all possible outcomes (events), ΩΩ For a coin toss, ΩΩ = {T,H} A rule for assigning values to X: X = ( 0 if T, 1 if H The sample space is a countable set of events (finite or infinite) 5
6 Discrete Random Variables Range / support x i 2{x 1,x 2,x n } Probability mass function P[X = x i ]=f(x i )=p i Cumulative distribution function P[X apple x i ]=F (x i )= ix j=1 p j 0 apple F (x i ) apple 1 6
7 Discrete Random Variables Question Consider a toss of a fair coin Define the random variable, X: ( 0 if T, X = 1 if H Write down the probability mass function and the (cumulative) distribution function to describe this probability model Answer Probability mass function: Cumulative distribution function: f(0) = 05 f(1) = 05 F (0) = 05 F (1) = 1 7
8 Discrete Random Variables Question Consider tossing two fair coins Define Y as the number of heads observed Write down the sample space for the experiment Determine the mass function and distribution function for Y Calculate the probability of observing at least one head Answer Sample space = {T,T}, {T,H}, {H, T}, {H, H} Support y 2 {0, 1, 2} 8
9 Discrete Random Variables Answer (continued) Probability functions y f(y) F (y) Probability of at least one head P[Y > 0] = 1 F (0) =1 025 =075 9
10 Continuous Random Variables Range / support (Probability) density function, f(x) 10 P[X = a] = Z a a f(x)dx =0 a b f P(a X b) x 2 [x l,x u ] x l <x u P[a <Xapple b] = Z b a f(x)dx x l <a<b<x u 0 < P[a <Xapple b] < 1 P[x l <Xapple x u ]= Z xu x l f(x)dx =1 P[a <X<b]=P[a apple X apple b]
11 Continuous Random Variables (Cumulative) distribution function, F(x) F (x) = Z x 1 f(u)du f(x) = df (x) dx Probability density function f(x) Cumulative distribution function 1 F(x) 0 x P[a <Xapple b] = Z b a f(x)dx = F (b) F (a) 11
12 Continuous Random Variables Question A continuous random variable, X, is uniformly distributed over the interval Determine the probability x 2 [0, 1] P[01 <Xapple 06] Answer Density function f(x) = ( 1 0 apple x apple 1, 0 otherwise 1 f (x) f(x) =1 x 2 [0, 1] A uniform distribution on [0, 1] interval x 12
13 Continuous Random Variables Answer (continued) Distribution function F(x) F (x) = Z x 0 f(u)du = x x 2 [0, 1] 1 (0, 0) 1 x Probability P[01 <Xapple 06] P[01 <Xapple 06] = F (06) F (01) = =05 13
14 Continuous Random Variables Question Consider the following distribution function F (x) =x 2 x 2 [0, 1] Find x 0 such that P[X apple x 0 ]= 1 4 Answer P[X apple x 0 ]= 1 4 F (x 0 )= 1 4 x 2 0 = 1 4 x 0 = r 1 4 x 0 =
15 Continuous Random Variables Answer (continued) Inverse distribution function F 1 F (x) = x If F F (x) =x 2 x = p F (x) 1 F (x) = p F (x) 1 F 1 = 4 r 1 4 = 1 2 For any probability, p F p = F (x) 1 (p) = p p = x 15
16 Joint Probability Functions Discrete random variables, X and Y x i 2 {x 1,,x n } y j 2 {y 1,,y m } Joint mass function P[X = x i \ Y = y j ]=f(x i,y j ) Joint distribution function P[X apple x k \ Y apple y l ]= kx lx f(x i,y j )=F(x k,y l ) i=1 j=1 16
17 Joint Probability Functions Continuous random variables, X and Y x 2 [x l,x u ] y 2 [y l,y u ] Joint density function f(x, y) Joint distribution function P[X apple x 0 \ Y apple y 0 ]= Z x0 x l Z y0 y l f(x, y)dy dx = F (x 0,y 0 ) 17
18 Moments A FINANCIAL TIMES COMPANY 18
19 Expectation Discrete random variables E[X] = nx x i f(x i )=µ x i 2 {x 1,x 2,x n } i=1 nx f(x i )=1 i=1 Continuous random variables E[X] = Z xu x l xf(x)dx = µ x 2 [x l,x u ] Z xu x l f(x)dx =1 19
20 Expectations of functions of random variables Expectation of a linear function of a random variable, X Y = a + bx E[Y ]=a + be[x] (a, b constants) Expectation of non-linear function of random variable, X Convex function Y = X 2 E[Y ] > E[X] 2 E[a + bx 2 ]=a + be[x 2 ] >a+ be[x] 2 Concave function Y = p X Jensen s Inequality E[Y ] < p E[X] 20
21 Moments Moments can be defined about any parameter (constant), c The n th moment about c, is E[(X c) n ] The n th ordinary moment of a random variable is a moment about zero (c =0) E[X n ] The first ordinary moment (n=1) is the mean of a random variable It is a measure of location or central tendency E[X] Moments defined about the mean of a random variable are central moments h E X E[X] ni = µ n 21
22 Moments The first (n = 1) central moment of a random variable is zero h i µ 1 = E X E[X] = E[X] E[X] = 0 The second central moment (n=2) of a random variable is variance By Jensen s inequality we know that the variance is always positive Variance is a measure of dispersion h µ 2 = E X E[X] 2i = E hx 2 2XE[X]+E[X] 2i = E[X 2 ] E[X] 2 = V[X] = 2 22
23 Variance Variance of a linear function of a random variable V[a + bx] =b 2 V[X] Variance of a non-linear function of a random variable Y = X 2 x f(x) E[X] =1 V[X] = 1 2 V[Y ]=E[Y 2 ] E[Y ] 2 = E[X 4 ] E[X 2 ] 2 = V[X 2 ] y f(y) E[Y ]= 3 2 V[Y ]= 9 4 V[X 2 ] 6= V[X] 2 V[a + bx 2 ]=b 2 V[X 2 ] 23
24 Standardized Moments The first standardized central moment of a random variable is zero µ 1 = h i E X E[X] p =0 2 The second standardized central moment of a random variable is one h µ E 2 2 = X E[X] 2i 2 =1 24
25 Skewness The third central moment of a random variable is a measure of the asymmetry of the density/mass function of the random variable h µ 3 = E X E[X] 3i = E[X 3 ] 3E[X]V[X] E[X] 3 = E[X 3 ] 3µ 2 µ 3 Symmetry is implied by h E X E[X] 3i =0 Skewness is defined as the third standardized central moment h S[X] = µ E 3 3 = X E[X] 3i 3 Probability density function Zero skewness Positive skewness Standard deviation = p V[X] Negative skewness ct of Skewness 25
26 Kurtosis The fourth central moment of a random variable is a measure of the dispersion of the distribution of the random variable h µ 4 = E X E[X] 4i Compared to the second central moment (variance), the fourth central moment places more weight on extreme values of the random variable Kurtosis is defined as the fourth standardized central moment of the random variable h K[X] = µ E 4 4 = X V[X] 2 E[X] 4i 26
27 Skewness and Kurtosis Question Determine the skewness and kurtosis for the random variable, X x f(x) # 04# 03# 02# 01# 00# f(x)% 0# 1# Answer E[X] = 0(05) + 1(05) = 05 V[X] =(0 05) 2 (05) + (1 05) 2 (05) = 025 S[X] = (0 05)3 (05) + (1 05) 3 (05) 05 3 =0 K[X] = (0 05)4 (05) + (1 05) 4 (05) 05 4 =1 Note: Kurtosis = 1 is the minimum kurtosis for any random variable! 27
28 Kurtosis Question Compare the kurtosis for the following random variables x f(x) # 050# 040# 030# 020# 010# f(x)% y f(y) # 060# 040# 020# f(y)% 000# 0# 1# 2# 000# 0# 1# 2# z f(z) # 060# 040# 020# 000# f(z)% (04142# 1# 24142# 28
29 Kurtosis Answer 422 x f(x) V[X] =(0 1) 2 (025) + (2 1) 2 (025) = 05 K[X] = (0 1)4 (025) + (2 1) 4 (025) 05 2 =2 y f(y) V[Y ]=(0 1) 2 (0125) + (2 1) 2 (0125) = 025 K[Y ]= (0 1)4 (0125) + (2 1) 4 (0125) =4 z f(z) V[Z] =( ) 2 (0125) + ( ) 2 (0125) 05 K[Z] = ( )4 (0125) + ( ) 4 (0125)
30 Kurtosis Kurtosis is best thought of as a measure of the heaviness of the tails of the density (mass) function of a continuous (discrete) random variable and of the peakedness of the density (mass) function The effects of kurtosis on tails and peaks across two or more distributions should only be compared if the variances are equal Excess Kurtosis for a selection of distributions with mean = 0 and variance = 1 Excess Kurtosis = Kurtosis - 3 Excess Kurtosis for a Normal RV = 0 Source: 30
31 Kurtosis Normal density: mean = 0, variance = 5/3, kurtosis = 3 Student-t density: mean = 0, deg of freedom = 5, variance = 5/3, kurtosis = 9 060# 050# 040# 030# 020# 010# 000# *700# *500# *300# *100# 100# 300# 500# 700# Normal density: mean = 0, variance = 1, kurtosis = 3 Student-t density: mean = 0, deg of freedom = 5, variance (scaled) = 1, kurtosis = 9 060# 050# 040# 030# 020# 010# 000# *700# *500# *300# *100# 100# 300# 500# 700# 31
32 Combinations of random variables Combinations Functions (sums, products, etc) of random variables Linear Combinations Moments Z = X + Y V = ax + by W = c + dx ey E[Z], V[Z], S[Z], K[Z] Joint probability (mass/density) f(x,y) y x
33 Combinations of random variables Expectation of linear combinations f(x,y) y x Z"="X"+"Y y x E[Z] y x E[Z] =165 f(x) x E[X] 085 f(y) y E[Y] 080 E[Z] =E[X]+E[Y ] =165 E[X + Y ]=E[X]+E[Y ] E[X Y ]=E[X] E[Y ] E[aX + by cz] = ae[x]+ be[y ] ce[z] 33
34 Combinations of random variables Variance of linear combinations V[Z] =E (Z E[Z]) 2 f(x,y) y x (z#$#e[z])^2 y x V[Z] =19275 Z^2$=$(X+Y)^2 y x V[Z] =E[Z 2 ] E[Z] 2 = =
35 Combinations of random variables Variance of linear combinations V[X + Y ]=E (X + Y E[X + Y ]) 2 = E[(X + Y ) 2 ] E[X + Y ] 2 = E[X 2 + Y 2 +2XY ] E[X] 2 + E[Y ] 2 +2E[X]E[Y ] = V[X]+V[Y ]+2 E[XY ] E[X]E[Y ] Covariance COV[X, Y ]=E[XY ] E[X]E[Y ] 35
36 Combinations of random variables Covariance A second order cross central moment h COV[X, Y ]=E X E[X] Y E[Y ] i = E[XY ] E[X]E[Y ] = X,Y y y y m Y m Y m Y x x m X m X m X x COV[X, Y ] > 0 COV[X, Y ] < 0 COV[X, Y ]=0 COV[X, X] =V[X] 36
37 Combinations of random variables Variance of linear combinations V[X + Y ]=E[(X + Y ) 2 ] E[X + Y ] 2 = V[X]+V[Y ]+2COV[X, Y ] Correlation V[aX + by ]=a 2 V[X]+b 2 V[Y ]+2ab COV[X, Y ] V[aX by ]=a 2 V[X]+b 2 V[Y ] 2ab COV[X, Y ] C[X, Y ]= COV[X, Y ] p V[X] V[Y ] = X,Y X Y = X,Y 1 apple X,Y apple 1 Expectation of products E[XY ]=E[X]E[Y ]+COV[X, Y ] Independence E[XY ]=E[X]E[Y ]! COV[X, Y ]=0 37
38 Probability Models A FINANCIAL TIMES COMPANY 38
39 Bernoulli Distribution Range (support) x 2 {0, 1} Mass function f(x) = ( 1 p if x =0 p if x =1 or f(x) =p x (1 p) 1 x 0 <p<1 Moments E[X] =p V[X] =p(1 p) S[X] = 1 2p p p(1 p) K[X] = 1 p(1 p) 3 f(x)% 05# 04# 03# 02# 01# 00# 0# 1# K(X) = 1, if p = 05! minimum kurtosis! 39
40 Binomial Distribution Sum of n independent and identically distributed (same p) Bernoulli random variables nx Y = i=1 X i Range (support) y 2 {0, 1, 2,,n} Mass function n f(y) = p y (1 y p) n y 0 <p<1 n = y n! y!(n y)! n! =n(n 1)(n 2) 1 Moments E[X] =np V[X] =np(1 p) 1 2p S[X] = p np(1 p) K[X] = 1 np(1 p) 6 n +3 Source: distribution 40
41 Binomial Distribution Question A multiple choice exam has 10 questions with 5 choices per question If you need at least 3 correct answers to pass the exam, what is the probability that you will pass simply by guessing? Answer Number of correct answers is a binomial random variable with n = 10 and p = 02 P[Y 3] = 1 P[Y apple 2] 10 = (08) (08) 9 0 = (004) + 10(016) = (098)
42 Poisson Distribution Range (support) x 2 {0, 1, 2,} Mass function f(x) = ( t)x x! e t t is a scale parameter Moments E[X] = V[X] = t t S[X] = 1 p t K[X] = 1 t +3 Source: 42
43 Poisson Distribution Question The number of defaults per month in a large bond portfolio follows a Poisson process On average, there are two defaults per month The number of defaults is independent from one month to the next Calculate the probability the number of defaults will not exceed 3 over the next two months Answer =2 t =2 P[X apple 3] = P[X = 0] + P[X = 1] + P[X = 2] + P[X = 3] = 40 0! e ! e ! e ! e 4 4 = e 4 0 0! ! ! ! = e = e
44 Poisson Approximation to Binomial Binomial probability P[Y = y] = n p y (1 p) n y y Poisson probability P[Y = y] =e y y! lim n!1 p!0 n p y (1 p) n y = e y y y! np = (constant) For large n and small p n p y (1 p) n y e y y y! np = 44
45 Exponential Distribution Range (support) t 2 [0, 1] Density function f(t) = e t Distribution function F (t) =1 e t Moments E(T )=1/ V(T )=1/ 2 S(T )=2 K(T )=9 Source: 45
46 Poisson and Exponential Distributions Question The number of defaults per month in a large bond portfolio follows a Poisson process On average, there are two defaults per month The number of defaults is independent from one month to the next Calculate the probability that there will be at least one default over the next month Answer Poisson: Exponential: =2 t =1 =2 P[X 1] = 1 P[X = 0] = ! e 2 =1 e P[T <1] = 1 e 2(1)
47 Uniform Distribution Range (support) x 2 [a, b] 1<a<b<1 Density function Distribution function Moments f(x) = 1 b F (x) = x b a a a E[X] = a + b 2 (b a)2 V[X] = 12 S[X] =0 K[X] = 9 5 Source: 47
48 Standard Normal (Gaussian) Distribution Range (support) z 2 ( 1, 1) Density function (z) = 1 p 2 e z2 /2 Distribution function 68% of the distribution is between 1 and +1 (z) = Z z 1 1 p 2 e u2 /2 du 95% is between 2 and +2 Moments Realization of the standard normal random variable E[Z] =0 V[Z] =1 S[Z] =0 K[Z] =3 Source: Jorion (2011), p 44 48
49 Normal (Gaussian) Distribution X = µ + Z Z N(0, 1) Z = X µ Range (support) Density function x 2 [ 1, 1] f(x) = 1 p 2 e 1 2( (x µ ) 2 Moments E[X] =µ V[X] = 2 S[X] =0 K[X] =3 The sum of independent normal random variables is a normal random variable Source: 49
50 Log-Normal Distribution X N(µ, ) Y = e X ln Y N(µ, ) Range (support) y 2 (0, 1) 14" 12" 10" 8" 6" 4" 2" Y"="exp(X)" Density function Moments f(y) = 1 y p 2 e 1 2( (ln y µ ) 2 0" (15" (10" (05" 00" 05" 10" 15" 20" 25" E[Y ]=e µ+ 2 /2 V[Y ]= e 2 1 e 2µ+ 2 p S[Y ]= e 2 +2 e 2 1 K[Y ]=e e e Source: 50
51 Chi-Squared Distribution X = kx Zi 2 Z i N(0, 1) iid X 2 (k) i=1 Range (support) x 2 [0, 1) Density function (a) = Z 1 0 f(x) = e y y a 1 2 k/2 k 2 1 dy x k/2 1 e x/2 (a) =(a 1)! (for integral values of a) Moments E[X] =k V[X] =2k S[X] = p 8/k K[X] = 12/k +3 Source: 51
52 Student s t Distribution X = Z p U/k Z N(0, 1) U 2 (k) Range (support) x 2 ( 1, 1) Density function f(x) = (a) = k+1 2 p k k 2 Z 1 0 e y y a 1+ x2 1 dy k k+1 2 Moments E[X] =0 k>1 V[X] =k/(k 2) k>2 S[X] =0 k>3 K[X] =6/(k 4) + 3 k>4 Source: 52
Random Variables Handout. Xavier Vilà
Random Variables Handout Xavier Vilà Course 2004-2005 1 Discrete Random Variables. 1.1 Introduction 1.1.1 Definition of Random Variable A random variable X is a function that maps each possible outcome
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationCS145: Probability & Computing
CS145: Probability & Computing Lecture 8: Variance of Sums, Cumulative Distribution, Continuous Variables Instructor: Eli Upfal Brown University Computer Science Figure credits: Bertsekas & Tsitsiklis,
More informationIEOR 165 Lecture 1 Probability Review
IEOR 165 Lecture 1 Probability Review 1 Definitions in Probability and Their Consequences 1.1 Defining Probability A probability space (Ω, F, P) consists of three elements: A sample space Ω is the set
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationProbability Distributions for Discrete RV
Probability Distributions for Discrete RV Probability Distributions for Discrete RV Definition The probability distribution or probability mass function (pmf) of a discrete rv is defined for every number
More informationCERTIFICATE IN FINANCE CQF. Certificate in Quantitative Finance Subtext t here GLOBAL STANDARD IN FINANCIAL ENGINEERING
CERTIFICATE IN FINANCE CQF Certificate in Quantitative Finance Subtext t here GLOBAL STANDARD IN FINANCIAL ENGINEERING Certificate in Quantitative Finance Probability and Statistics June 2011 1 1 PROBABILITY
More informationFV N = PV (1+ r) N. FV N = PVe rs * N 2011 ELAN GUIDES 3. The Future Value of a Single Cash Flow. The Present Value of a Single Cash Flow
QUANTITATIVE METHODS The Future Value of a Single Cash Flow FV N = PV (1+ r) N The Present Value of a Single Cash Flow PV = FV (1+ r) N PV Annuity Due = PVOrdinary Annuity (1 + r) FV Annuity Due = FVOrdinary
More informationDiscrete Random Variables and Probability Distributions
Chapter 4 Discrete Random Variables and Probability Distributions 4.1 Random Variables A quantity resulting from an experiment that, by chance, can assume different values. A random variable is a variable
More informationProbability Theory and Simulation Methods. April 9th, Lecture 20: Special distributions
April 9th, 2018 Lecture 20: Special distributions Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters 4, 6: Random variables Week 9 Chapter
More informationEcon 250 Fall Due at November 16. Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling
Econ 250 Fall 2010 Due at November 16 Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling 1. Suppose a firm wishes to raise funds and there are a large number of independent financial
More informationECO220Y Introduction to Probability Readings: Chapter 6 (skip section 6.9) and Chapter 9 (section )
ECO220Y Introduction to Probability Readings: Chapter 6 (skip section 6.9) and Chapter 9 (section 9.1-9.3) Fall 2011 Lecture 6 Part 2 (Fall 2011) Introduction to Probability Lecture 6 Part 2 1 / 44 From
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationLecture 2. Probability Distributions Theophanis Tsandilas
Lecture 2 Probability Distributions Theophanis Tsandilas Comment on measures of dispersion Why do common measures of dispersion (variance and standard deviation) use sums of squares: nx (x i ˆµ) 2 i=1
More informationTopic 6 - Continuous Distributions I. Discrete RVs. Probability Density. Continuous RVs. Background Reading. Recall the discrete distributions
Topic 6 - Continuous Distributions I Discrete RVs Recall the discrete distributions STAT 511 Professor Bruce Craig Binomial - X= number of successes (x =, 1,...,n) Geometric - X= number of trials (x =,...)
More information15.063: Communicating with Data Summer Recitation 3 Probability II
15.063: Communicating with Data Summer 2003 Recitation 3 Probability II Today s Goal Binomial Random Variables (RV) Covariance and Correlation Sums of RV Normal RV 15.063, Summer '03 2 Random Variables
More informationReview for Final Exam Spring 2014 Jeremy Orloff and Jonathan Bloom
Review for Final Exam 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom THANK YOU!!!! JON!! PETER!! RUTHI!! ERIKA!! ALL OF YOU!!!! Probability Counting Sets Inclusion-exclusion principle Rule of product
More informationExpectations. Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or
Definition Let X be a discrete rv with set of possible values D and pmf p(x). The expected value or mean value of X, denoted by E(X ) or µ X, is E(X ) = µ X = x D x p(x) Definition Let X be a discrete
More informationRandom variables. Contents
Random variables Contents 1 Random Variable 2 1.1 Discrete Random Variable............................ 3 1.2 Continuous Random Variable........................... 5 1.3 Measures of Location...............................
More informationSTAT Chapter 4/6: Random Variables and Probability Distributions
STAT 251 - Chapter 4/6: Random Variables and Probability Distributions We use random variables (RV) to represent the numerical features of a random experiment. In chapter 3, we defined a random experiment
More informationStatistics for Managers Using Microsoft Excel 7 th Edition
Statistics for Managers Using Microsoft Excel 7 th Edition Chapter 5 Discrete Probability Distributions Statistics for Managers Using Microsoft Excel 7e Copyright 014 Pearson Education, Inc. Chap 5-1 Learning
More informationDiscrete Random Variables and Probability Distributions. Stat 4570/5570 Based on Devore s book (Ed 8)
3 Discrete Random Variables and Probability Distributions Stat 4570/5570 Based on Devore s book (Ed 8) Random Variables We can associate each single outcome of an experiment with a real number: We refer
More informationChapter 2. Random variables. 2.3 Expectation
Random processes - Chapter 2. Random variables 1 Random processes Chapter 2. Random variables 2.3 Expectation 2.3 Expectation Random processes - Chapter 2. Random variables 2 Among the parameters representing
More informationContinuous random variables
Continuous random variables probability density function (f(x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable),
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 28 One more
More informationWelcome to Stat 410!
Welcome to Stat 410! Personnel Instructor: Liang, Feng TA: Gan, Gary (Lingrui) Instructors/TAs from two other sessions Websites: Piazza and Compass Homework When, where and how to submit your homework
More informationContents. An Overview of Statistical Applications CHAPTER 1. Contents (ix) Preface... (vii)
Contents (ix) Contents Preface... (vii) CHAPTER 1 An Overview of Statistical Applications 1.1 Introduction... 1 1. Probability Functions and Statistics... 1..1 Discrete versus Continuous Functions... 1..
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More information1/2 2. Mean & variance. Mean & standard deviation
Question # 1 of 10 ( Start time: 09:46:03 PM ) Total Marks: 1 The probability distribution of X is given below. x: 0 1 2 3 4 p(x): 0.73? 0.06 0.04 0.01 What is the value of missing probability? 0.54 0.16
More informationCommonly Used Distributions
Chapter 4: Commonly Used Distributions 1 Introduction Statistical inference involves drawing a sample from a population and analyzing the sample data to learn about the population. We often have some knowledge
More informationChapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions Part 2: More on Continuous Random Variables Section 4.5 Continuous Uniform Distribution Section 4.6 Normal Distribution 1 / 27 Continuous
More informationWeek 1 Quantitative Analysis of Financial Markets Basic Statistics A
Week 1 Quantitative Analysis of Financial Markets Basic Statistics A Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More informationReliability and Risk Analysis. Survival and Reliability Function
Reliability and Risk Analysis Survival function We consider a non-negative random variable X which indicates the waiting time for the risk event (eg failure of the monitored equipment, etc.). The probability
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Moments of a distribubon Measures of
More informationModel Paper Statistics Objective. Paper Code Time Allowed: 20 minutes
Model Paper Statistics Objective Intermediate Part I (11 th Class) Examination Session 2012-2013 and onward Total marks: 17 Paper Code Time Allowed: 20 minutes Note:- You have four choices for each objective
More informationME3620. Theory of Engineering Experimentation. Spring Chapter III. Random Variables and Probability Distributions.
ME3620 Theory of Engineering Experimentation Chapter III. Random Variables and Probability Distributions Chapter III 1 3.2 Random Variables In an experiment, a measurement is usually denoted by a variable
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 207 Homework 5 Drew Armstrong Problems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman: Section 3., Exercises 3, 0. Section 3.3, Exercises 2, 3, 0,.
More informationExam P September 2014 Study Sheet! copylefted by Jared Nakamura, 9/4/2014!
Exam P September 2014 Study Sheet copylefted by Jared Nakamura, 9/4/2014 I. General Probability (15-30%) A. Set functions including set notation and basic elements of probability 1. Set notation: A = {a1,
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationSTOR Lecture 7. Random Variables - I
STOR 435.001 Lecture 7 Random Variables - I Shankar Bhamidi UNC Chapel Hill 1 / 31 Example 1a: Suppose that our experiment consists of tossing 3 fair coins. Let Y denote the number of heads that appear.
More informationcontinuous rv Note for a legitimate pdf, we have f (x) 0 and f (x)dx = 1. For a continuous rv, P(X = c) = c f (x)dx = 0, hence
continuous rv Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f(x) such that for any two numbers a and b with a b, P(a X b) = b a f (x)dx.
More informationBinomial Random Variables. Binomial Random Variables
Bernoulli Trials Definition A Bernoulli trial is a random experiment in which there are only two possible outcomes - success and failure. 1 Tossing a coin and considering heads as success and tails as
More informationStatistics for Business and Economics: Random Variables:Continuous
Statistics for Business and Economics: Random Variables:Continuous STT 315: Section 107 Acknowledgement: I d like to thank Dr. Ashoke Sinha for allowing me to use and edit the slides. Murray Bourne (interactive
More informationMATH MW Elementary Probability Course Notes Part IV: Binomial/Normal distributions Mean and Variance
MATH 2030 3.00MW Elementary Probability Course Notes Part IV: Binomial/Normal distributions Mean and Variance Tom Salisbury salt@yorku.ca York University, Dept. of Mathematics and Statistics Original version
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationChapter 7. Sampling Distributions and the Central Limit Theorem
Chapter 7. Sampling Distributions and the Central Limit Theorem 1 Introduction 2 Sampling Distributions related to the normal distribution 3 The central limit theorem 4 The normal approximation to binomial
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationStatistics 6 th Edition
Statistics 6 th Edition Chapter 5 Discrete Probability Distributions Chap 5-1 Definitions Random Variables Random Variables Discrete Random Variable Continuous Random Variable Ch. 5 Ch. 6 Chap 5-2 Discrete
More information6 If and then. (a) 0.6 (b) 0.9 (c) 2 (d) Which of these numbers can be a value of probability distribution of a discrete random variable
1. A number between 0 and 1 that is use to measure uncertainty is called: (a) Random variable (b) Trial (c) Simple event (d) Probability 2. Probability can be expressed as: (a) Rational (b) Fraction (c)
More informationThe Bernoulli distribution
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationCovariance and Correlation. Def: If X and Y are JDRVs with finite means and variances, then. Example Sampling
Definitions Properties E(X) µ X Transformations Linearity Monotonicity Expectation Chapter 7 xdf X (x). Expectation Independence Recall: µ X minimizes E[(X c) ] w.r.t. c. The Prediction Problem The Problem:
More informationStatistical Methods for NLP LT 2202
LT 2202 Lecture 3 Random variables January 26, 2012 Recap of lecture 2 Basic laws of probability: 0 P(A) 1 for every event A. P(Ω) = 1 P(A B) = P(A) + P(B) if A and B disjoint Conditional probability:
More informationChapter 7. Sampling Distributions and the Central Limit Theorem
Chapter 7. Sampling Distributions and the Central Limit Theorem 1 Introduction 2 Sampling Distributions related to the normal distribution 3 The central limit theorem 4 The normal approximation to binomial
More informationPROBABILITY AND STATISTICS
Monday, January 12, 2015 1 PROBABILITY AND STATISTICS Zhenyu Ye January 12, 2015 Monday, January 12, 2015 2 References Ch10 of Experiments in Modern Physics by Melissinos. Particle Physics Data Group Review
More informationTOPIC: PROBABILITY DISTRIBUTIONS
TOPIC: PROBABILITY DISTRIBUTIONS There are two types of random variables: A Discrete random variable can take on only specified, distinct values. A Continuous random variable can take on any value within
More informationRandom Variables and Probability Functions
University of Central Arkansas Random Variables and Probability Functions Directory Table of Contents. Begin Article. Stephen R. Addison Copyright c 001 saddison@mailaps.org Last Revision Date: February
More informationProbability Distributions II
Probability Distributions II Summer 2017 Summer Institutes 63 Multinomial Distribution - Motivation Suppose we modified assumption (1) of the binomial distribution to allow for more than two outcomes.
More information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
More informationECON Introductory Econometrics. Lecture 1: Introduction and Review of Statistics
ECON4150 - Introductory Econometrics Lecture 1: Introduction and Review of Statistics Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 1-2 Lecture outline 2 What is econometrics? Course
More informationStatistics. Marco Caserta IE University. Stats 1 / 56
Statistics Marco Caserta marco.caserta@ie.edu IE University Stats 1 / 56 1 Random variables 2 Binomial distribution 3 Poisson distribution 4 Hypergeometric Distribution 5 Jointly Distributed Discrete Random
More informationECE 340 Probabilistic Methods in Engineering M/W 3-4:15. Lecture 10: Continuous RV Families. Prof. Vince Calhoun
ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Lecture 10: Continuous RV Families Prof. Vince Calhoun 1 Reading This class: Section 4.4-4.5 Next class: Section 4.6-4.7 2 Homework 3.9, 3.49, 4.5,
More informationCentral Limit Theorem (cont d) 7/28/2006
Central Limit Theorem (cont d) 7/28/2006 Central Limit Theorem for Binomial Distributions Theorem. For the binomial distribution b(n, p, j) we have lim npq b(n, p, np + x npq ) = φ(x), n where φ(x) is
More informationSimulation Wrap-up, Statistics COS 323
Simulation Wrap-up, Statistics COS 323 Today Simulation Re-cap Statistics Variance and confidence intervals for simulations Simulation wrap-up FYI: No class or office hours Thursday Simulation wrap-up
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationChapter 3 Discrete Random Variables and Probability Distributions
Chapter 3 Discrete Random Variables and Probability Distributions Part 3: Special Discrete Random Variable Distributions Section 3.5 Discrete Uniform Section 3.6 Bernoulli and Binomial Others sections
More information2 of PU_2015_375 Which of the following measures is more flexible when compared to other measures?
PU M Sc Statistics 1 of 100 194 PU_2015_375 The population census period in India is for every:- quarterly Quinqennial year biannual Decennial year 2 of 100 105 PU_2015_375 Which of the following measures
More informationChapter 7: Random Variables
Chapter 7: Random Variables 7.1 Discrete and Continuous Random Variables 7.2 Means and Variances of Random Variables 1 Introduction A random variable is a function that associates a unique numerical value
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationModule 3: Sampling Distributions and the CLT Statistics (OA3102)
Module 3: Sampling Distributions and the CLT Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chpt 7.1-7.3, 7.5 Revision: 1-12 1 Goals for
More information5. In fact, any function of a random variable is also a random variable
Random Variables - Class 11 October 14, 2012 Debdeep Pati 1 Random variables 1.1 Expectation of a function of a random variable 1. Expectation of a function of a random variable 2. We know E(X) = x xp(x)
More informationA random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon.
Chapter 14: random variables p394 A random variable (r. v.) is a variable whose value is a numerical outcome of a random phenomenon. Consider the experiment of tossing a coin. Define a random variable
More informationConverting to the Standard Normal rv: Exponential PDF and CDF for x 0 Chapter 7: expected value of x
Key Formula Sheet ASU ECN 22 ASWCC Chapter : no key formulas Chapter 2: Relative Frequency=freq of the class/n Approx Class Width: =(largest value-smallest value) /number of classes Chapter 3: sample and
More informationSYLLABUS AND SAMPLE QUESTIONS FOR MSQE (Program Code: MQEK and MQED) Syllabus for PEA (Mathematics), 2013
SYLLABUS AND SAMPLE QUESTIONS FOR MSQE (Program Code: MQEK and MQED) 2013 Syllabus for PEA (Mathematics), 2013 Algebra: Binomial Theorem, AP, GP, HP, Exponential, Logarithmic Series, Sequence, Permutations
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
More informationIntroduction to Probability
August 18, 2006 Contents Outline. Contents.. Contents.. Contents.. Application: 1-d diffusion Definition Outline Consider M discrete events x = x i, i = 1,2,,M. The probability for the occurrence of x
More informationQuantitative Analysis
EduPristine www.edupristine.com/ca Future value Value of current cash flow in Future Compounding Present value Present value of future cash flow Discounting Annuities Series of equal cash flows occurring
More informationFavorite Distributions
Favorite Distributions Binomial, Poisson and Normal Here we consider 3 favorite distributions in statistics: Binomial, discovered by James Bernoulli in 1700 Poisson, a limiting form of the Binomial, found
More informationCS134: Networks Spring Random Variables and Independence. 1.2 Probability Distribution Function (PDF) Number of heads Probability 2 0.
CS134: Networks Spring 2017 Prof. Yaron Singer Section 0 1 Probability 1.1 Random Variables and Independence A real-valued random variable is a variable that can take each of a set of possible values in
More informationNormal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is
Normal Distribution Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More informationSTA 103: Final Exam. Print clearly on this exam. Only correct solutions that can be read will be given credit.
STA 103: Final Exam June 26, 2008 Name: } {{ } by writing my name i swear by the honor code Read all of the following information before starting the exam: Print clearly on this exam. Only correct solutions
More information1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by. Cov(X, Y ) = E(X E(X))(Y E(Y ))
Correlation & Estimation - Class 7 January 28, 2014 Debdeep Pati Association between two variables 1. Covariance between two variables X and Y is denoted by Cov(X, Y) and defined by Cov(X, Y ) = E(X E(X))(Y
More informationChapter 7: Random Variables and Discrete Probability Distributions
Chapter 7: Random Variables and Discrete Probability Distributions 7. Random Variables and Probability Distributions This section introduced the concept of a random variable, which assigns a numerical
More information1. If four dice are rolled, what is the probability of obtaining two identical odd numbers and two identical even numbers?
1 451/551 - Final Review Problems 1 Probability by Sample Points 1. If four dice are rolled, what is the probability of obtaining two identical odd numbers and two identical even numbers? 2. A box contains
More informationReview of the Topics for Midterm I
Review of the Topics for Midterm I STA 100 Lecture 9 I. Introduction The objective of statistics is to make inferences about a population based on information contained in a sample. A population is the
More information15.063: Communicating with Data Summer Recitation 4 Probability III
15.063: Communicating with Data Summer 2003 Recitation 4 Probability III Today s Content Normal RV Central Limit Theorem (CLT) Statistical Sampling 15.063, Summer '03 2 Normal Distribution Any normal RV
More information6. Continous Distributions
6. Continous Distributions Chris Piech and Mehran Sahami May 17 So far, all random variables we have seen have been discrete. In all the cases we have seen in CS19 this meant that our RVs could only take
More information2.1 Probability, stochastic variables and distribution functions
Chapter 2 Probability and statistics 2.1 Probability, stochastic variables and distribution functions The defining characteristic of a stochastic experiment E is that it produces different outcomes under
More informationMTP_Foundation_Syllabus 2012_June2016_Set 1
Paper- 4: FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS Academics Department, The Institute of Cost Accountants of India (Statutory Body under an Act of Parliament) Page 1 Paper- 4: FUNDAMENTALS
More informationMA : Introductory Probability
MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320:
More informationChapter 3 - Lecture 5 The Binomial Probability Distribution
Chapter 3 - Lecture 5 The Binomial Probability October 12th, 2009 Experiment Examples Moments and moment generating function of a Binomial Random Variable Outline Experiment Examples A binomial experiment
More informationCHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS
CHAPTER 7 RANDOM VARIABLES AND DISCRETE PROBABILTY DISTRIBUTIONS MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, please circle the correct answer.. The weighted average of the possible
More informationLecture 23. STAT 225 Introduction to Probability Models April 4, Whitney Huang Purdue University. Normal approximation to Binomial
Lecture 23 STAT 225 Introduction to Probability Models April 4, 2014 approximation Whitney Huang Purdue University 23.1 Agenda 1 approximation 2 approximation 23.2 Characteristics of the random variable:
More informationCambridge University Press Risk Modelling in General Insurance: From Principles to Practice Roger J. Gray and Susan M.
adjustment coefficient, 272 and Cramér Lundberg approximation, 302 existence, 279 and Lundberg s inequality, 272 numerical methods for, 303 properties, 272 and reinsurance (case study), 348 statistical
More informationExam M Fall 2005 PRELIMINARY ANSWER KEY
Exam M Fall 005 PRELIMINARY ANSWER KEY Question # Answer Question # Answer 1 C 1 E C B 3 C 3 E 4 D 4 E 5 C 5 C 6 B 6 E 7 A 7 E 8 D 8 D 9 B 9 A 10 A 30 D 11 A 31 A 1 A 3 A 13 D 33 B 14 C 34 C 15 A 35 A
More informationS t d with probability (1 p), where
Stochastic Calculus Week 3 Topics: Towards Black-Scholes Stochastic Processes Brownian Motion Conditional Expectations Continuous-time Martingales Towards Black Scholes Suppose again that S t+δt equals
More informationThe Normal Distribution
Will Monroe CS 09 The Normal Distribution Lecture Notes # July 9, 207 Based on a chapter by Chris Piech The single most important random variable type is the normal a.k.a. Gaussian) random variable, parametrized
More informationCase Study: Heavy-Tailed Distribution and Reinsurance Rate-making
Case Study: Heavy-Tailed Distribution and Reinsurance Rate-making May 30, 2016 The purpose of this case study is to give a brief introduction to a heavy-tailed distribution and its distinct behaviors in
More information