HUDM4122 Probability and Statistical Inference February 23, 2015
In the last class We studied Bayes Theorem and the Law of Total Probability
Any questions or comments?
Today Chapter 4.8 in Mendenhall, Beaver, & Beaver Probability Distributions Random Variables
Random Variable A variable x is a random variable if the value that it assumes, corresponding to the outcome of an experiment, is a chance or random event. MBB, p.158 A random variable is a variable whose value is subject to variations due to chance. -- Wikipedia
Random Variable It is not that the variable can have any value at random But that the variable s value comes from random sampling
These are random variables A coin flip s result Number of times a randomly selected student is sent to the principal s office NY Regents Exam score for a randomly selected student in NY State Number of people on the subway at a randomly selected time
These are not random variables I flip a biased coin that gives 100% heads The temperature setting on your oven you set it yourself These values are not subject to chance
A random variable s value You can never say for sure what it s going to be There is a certainly probability that it will have certain values
Questions? Comments?
Reminder Discrete variable Can have limited number of values Continuous/numerical variable Can have infinite number of values
Which of these are Discrete Variables? Number of heads in 3 coin flips Sum of rolling two 6-sided dice Temperature outside How late is 2 train Height of a person, rounded to closest inch Number of times a randomly selected student is sent to the principal s office
Probability Distribution A probability distribution for random variable X gives the possible values of X, x 1 x n And the probability p(x i ) associated with each value of X
Probability Distribution A probability distribution for random variable X gives the possible values of X, x 1 x n And the probability p(x i ) associated with each value of X Each value of X is mutually exclusive The sum of p(x i ) adds to 1
Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2
Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x 0 1 2 P(x)
Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x) 0 1/4 1 2/4 2 1/4
Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x) 0 1/4 1 1/2 2 1/4
Example I flip a coin twice The number of heads can be 0, 1, or 2 TT: 0 TH:1 HT:1 HH:2 x P(x) 0 0.25 1 0.5 2 0.25
Probability Histogram x P(x) 0.6 0 0.25 1 0.5 2 0.25 p(x) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 x
You try it I flip a coin three times What is the probability distribution on the number of heads?
You try it: What is the Probability Distribution? I collected the following data on the temperature in NYC Day Temp Day Temp 1 Cold 9 Cold 2 Cold 10 Cold 3 Freezing 11 Cold 4 Cold 12 Not That Bad 5 Cold 13 Cold 6 Freezing 14 F ing Freezing 7 F ing Freezing 15 F ing Freezing 8 Freezing 16 Cold
Note that probability distributions can have many values p(x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x
Later in the semester We ll talk about probability distributions for continuous variables
Questions? Comments?
Expected Value of Probability Distribution The value you can expect to get on average if you re-run your experiment many times Take each value, multiply it by its probability Add those together E(x) = ( )
Expected Value of Probability Distribution The value you can expect to get on average if you re-run your experiment many times Take each value, multiply it by its probability Add those together E(x) = ( ) This is also called the mean of the random variable, µ
Example x P(x) 0 0.25 1 0.5 2 0.25 0(0.25) + 1(0.5) + 2(0.25)
Example x P(x) 0 0.25 1 0.5 2 0.25 0 + 0.5 + 0.5 = 1
You can expect 1 head on average if you flip a coin twice, infinite times x P(x) 0 0.25 1 0.5 2 0.25 0 + 0.5 + 0.5 = 1
You Try It x P(x) 0 0.1 1 0.5 2 0.2 3 0.2
You Try It x P(x) 0 0.1 1 0.5 2 0.2 3 0.2 0(0.1)+1(0.5)+2(0.2)+3(0.2)
You Try It x P(x) 0+0.5+0.4+0.6= 1.5 0 0.1 1 0.5 2 0.2 3 0.2
You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery?
You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery? (1/14600)(1000) + (14599/14600)(-1)
You Try It My dad s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket One time he won $1000 What is the expected value for playing the lottery? (1/14600)(1000) + (14599/14600)(-1)= 0.068 0. 999 = $-0.931
Questions? Comments?
Variance of Discrete Random Variable ( )
Standard Deviation of Discrete Random Variable ( )
Example: SD of Random Variable x P(x) 0 0.25 1 0.5 2 0.25
Example: SD of Random Variable x P(x) Recall: µ = 1 0 0.25 1 0.5 2 0.25
Example: xsd of Random P(x) Variable 0 0.25 1 0.5 2 0.25 Recall: µ = 1 ( )
Example: xsd of Random P(x) Variable 0 0.25 1 0.5 2 0.25 Recall: µ = 1 0 1 (0.25) + 1 1.5 + 2 1.25
Example: xsd of Random P(x) Variable 0 0.25 1 0.5 2 0.25 Recall: µ = 1 1(0.25) + 0.5 +1.25
Example: xsd of Random P(x) Variable 0 0.25 1 0.5 2 0.25 Recall: µ = 1 0.25 + 0.25
Example: xsd of Random P(x) Variable 0 0.25 1 0.5 2 0.25 Recall: µ = 1 0.25 + 0.25 0.707
You Try It: SD of random variable x P(x) 0 0.1 1 0.5 2 0.2 3 0.2 µ = 1.5
Any last comments or questions for the day?
If there s time: Do this in solver-explainer pairs Practice with extended version of Bayes Rule ( ) = ( ) )
Example There are three professors who teach HUDM4122. Let s call them A, B, and C P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1
Example P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1 P(learned stats A)=0.8 P(learned stats B)=0.6 P(learned stats C)=0.4
What is P(A learned stats)? P(student is in prof A s class) = P(A) = 0.4 P(student is in prof B s class) = P(B) = 0.3 P(student is in prof C s class) = P(C) = 0.1 P(learned stats A)=0.8 P(learned stats B)=0.6 P(learned stats C)=0.4
Upcoming Classes 2/25 Binomial Probability Distribution Ch. 5-2 HW 4 due 2/28 Normal Probability Distribution
Homework 4 Due in 2 days In the ASSISTments system