Lecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.

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1 STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called rior robabilities; B A) is called the osterior robability of B. Examle: Consider a test for an illness. The test has a known reliability: (1). When administered to an ill erson, the test will indicate so with robability 0.9. (). When administered to a erson who is not ill, the test will erroneously give a ositive result with robability Suose the illness is rare and is known to affect only 0.1% of the entire oulation. If a erson is randomly selected from the entire oulation and is given the test and the result is ositive, what is the osterior robability that the erson is ill? Chater 3 Random Variables 1. Random Variable Examles: (1). Tossing a fair coin three times, let be the number of heads showed u. (). Let Y be the number of car accidents at East Lansing on May. (3). The bus will arrive at a bus sto at anytime between 10:00am and 1:00am. A student is waiting for the bus from 10:00am at the bus sto, Let T be the waiting time.

2 STT315, Section 701, Summer 006 Definition: A random variable is a function of the samle sace. Examle: Tossing a fair coin three times, let be the number of heads showed u. Samle Sace Random Variable H H H H H T H T H T H H T T H T H T H T T T T T = 3 = = 1 = 0 In the following, we will use the uercase letter, Y etc. to denote the random variable, and the lowercase letter x, y etc. to denote the articular value that the random variable can take. Discrete Random Variable: A discrete random variable can assume at most a countable number of values. Examles: Continuous Random Variable: A continuous random variable may take on any value in an interval of numbers. Examles: Probability Distribution of a Random Variable: Probability Distribution of a Discrete Random Variable: The robability distribution of a discrete variable can be reresented by a formula, a table, or a grah, which rovides (x)==x) for all x.

3 STT315, Section 701, Summer 006 Examle: Probability distribution of the Sum of Two Dice x (x) 1/36 3 /36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/ /36 11 /36 1 1/36 The robability distribution of a discrete random variable must satisfy the following two conditions. (1). (x) 0 for all value x; (). ( x) = 1. all x Cumulative Distribution Function: The cumulative distribution function F(x) of a discrete random variable is F ( x) = x) = ( y) all y x

4 STT315, Section 701, Summer 006. Exected Values of Discrete Random Variable (1). The exected value of a discrete random variable is equal to the sum of all values of the random variable, each value multilied by its robability. μ = E( ) = x( x) Examle: The robability distribution of the random variable is the following. Find the exected value of. all x x (x) (). The exected value of h(), a function of the discrete random variable is E[ h( )] = h( x) ( x). all x In articular, if h() = a + b, then E[h()] = a E() + b. Examle: 3. Variance and Standard Deviation of a Random Variable. (1) The variance of a discrete random variable is given by σ = V ( ) = E[( μ) ] = ( x μ) ( x). Also, V ( ) = E( all x ) [ E( )]. Examle:

5 STT315, Section 701, Summer 006 () Standard Deviation: The standard deviation of a random variable is given by σ = SD ( ) = V ( ) Examle: (3) Variance of a Linear Function of a Random Variable: V = ( a + b) = a V ( ) a σ 4. Sum and Linear Comosites of Random Variables A linear comosite of random variables 1,, L k will be of the form a a + L+ a k k The exected value of a linear comosite is given by E a + a + L + a ) = a E( ) + a E( ) + L+ a ( 1 1 k k 1 1 k E( k ) If 1,, L k are mutually indeendent, then V ( a + a + L + a ) = a E( ) + a V ( ) + L+ a V ( 1 1 k k 1 1 k k ) 5. Some Discrete Probability Distributions. (1). Bernoulli Random Variable: If the outcome of a trial can only be either a success or a failure, then the trial is a Bernoulli trial. Let be the number of successes in one Bernoulli trial, which can be 1 or 0, is a Bernoulli random variable. Notation ~ BER(). The robability distribution of a Bernoulli random variable is Clearly, E() =, V() = (1-). x (x)

6 STT315, Section 701, Summer 006 (). Binomial Probability Distribution. Binomial Exeriment: A binominal exeriment ossesses the following roerties i. The exeriment consists of a fixed number, n, of identical Bernoulli trials. ii. The robability of success is and the robability of a failure is equal to q = 1-. iii. The trials are indeendent. iv. The random variable of interest is, the number of successes observed during the n trials. Examle: An early-warning detection system for aircraft consists of four identical radar units oerating indeendently of one another. Suose that each has a robability 0.95 of detecting an intruding aircraft. When an intruding aircraft enters the scene, the random variable of interest is, the number of radar units that do not detect the lane. Is this a binomial exeriment? Examle: Suose that 40% of a large oulation of registered voters favor candidate Jones. A random samle of n =10 voters will be selected, and, the number of favoring Jones, is to be observed. Does this exeriment meet the requirements of a binomial exeriment. Probability Distribution of a Binomial Random Variable. Let be the number of successes in a Binomial exeriment, then is called a Binomial random variable, the robability distribution is: Notation: ~ B(n, ). n x x x x q n ( ) =, x= 0, 1,,, n and 0 1. Examle: Exerience has shown that 30% of all ersons afflicted by a certain illness recover. A drug comany has develoed a new medication. Ten eole with the illness were selected at random and received the medication; nine recovered shortly thereafter. Suose that the medication was absolutely worthless. What is the robability that at least nine of ten receiving the medication will recover? Exectation and Variance of the Binomial Random Variable. Let ~ B(n, ). Then E () = n, V() = nq.

7 STT315, Section 701, Summer 006 (3). Negative Binomial Probability Distribution Consider the case of oerator who wants to roduce two good ins using a lathe that has 0.6 robability of making one good in in each trial. Suose two good ins are needed, the oerator would roduce the ins one by one and sto when he gets two good ones. Notice that in this scenario, the number of successes is held constant at, and the number of trials is random, which is called a negative binomial random variable. Generally, let s denote the exact number of successes desired and the robability of success in each trial. Let denote the number of trials made until the desired number of success is achieved. Then will follow a negative binomial distribution and we shall write ~ NB(s,). If ~ NB(s,), then x 1 = x) = s 1 s (1 ) x s. Exectation and Variance of the Negative Binomial Random Variable Let ~ NB(s,) s s(1 ) E ( ) =, V ( ) =. (4). Geometric Probability Distribution An exeriment involves identical and indeendent trials, each of which can result in one of two outcomes, success and failure. The robability of success is equal to and is constant from trial to trial. Let be the number of the trial on which the first success occurs. Then is called a geometric random variable. If is a geometric random variable, is the robability of success, then x = x) = q 1, x = 0, 1,,, n and 0 1 Examle: Suose that the robability of engine malfunction during any 1-hour eriod is =0.0. Find the robability that a given engine will survive hours.

8 STT315, Section 701, Summer 006 Exectation and Variance of the Geometric Random Variable Let be a geometric random variable. 1 1 E ( ) =, V ( ) = (5). Hyergeometric Probability Distribution Suose that a oulation contains a finite number N of elements that ossess one of two characteristics. Thus S of the elements might be red and b=n-s, black. A samle of n elements is randomly selected from the oulation and the random variable of interest is, the number of red elements in the samle. Then is a hyergeometric random variable. If is a hyergeometric random variable, S N S x n x P ( = x) = N n Where x is an integer 0, 1,,, n, subject to the restriction x S and n-x N-S Examle: An imortant roblem encountered by ersonnel directors and others faced with the selection of the best in a finite set of elements is exemlified by the following scenario. From a grou of 0 Ph.D. engineers, 10 are randomly selected for emloyment. What is the robability that the 10 selected include all the 5 best engineers in the grou of 0.

9 STT315, Section 701, Summer 006 Exectation and Variance of the Hyergeometric Random Variable Let be a hyergeometric random variable. ns S N S N n E ( ) =, V ( ) = n N N N N 1 (6). Poisson Probability Distribution Derivation of Poisson Probability Distribution from Binomial Probability Distribution. Suose that we want to find the robability distribution of the number of automobile accidents at a articular intersection during a time eriod of one week. Think of the time eriod, one week in this examle, as being slit u into n subintervals, each of which is so small that at most one accident could occur in it with robability different from 0. Denoting the robability of an accident in any subinterval by, we have one accident occurs in a subinterval) =, no accidents occur in a subinterval) = 1-. Assume decreases when n increases, and n = l. Then x n x n x λ λ lim (1 ) = e. n x x! If the distribution of a random variable has the following robability distribution, x λ λ = x) = e, x = 0, 1,, x! Then is called a Poisson Random Variable, denoted by ~ l).

10 STT315, Section 701, Summer 006 Examle: Suose that a random system of olice atrol is devised so that a atrol officer may visit a given beat location = 0, 1,, times er half-hour eriod, with each location being visited an average of once er time eriod. Assume ossesses, aroximately, a Poisson robability distribution. Calculate the robability that the atrol officer will miss a given location during a half-hour eriod. What is the robability that it will be visited once? Twice? At least once? Exectation and Variance of the Possion Random Variable Let be a Possion random variable with ~ l). E ( ) = λ, V ( ) = λ. 6. Some Continuous Random Variable Probability Density Function: A function, denoted by f(x), is called a robability density function, if it has the following roerties a) f(x) 0 for all x. b) The robability that will be between two numbers a and b is equal to the area under f(x) between a and b. c) The total area under the entire curve of f(x) is equal to 1. Cumulative Distribution Function: The cumulative distribution function of a continuous random variable F(x) = x ) = area under f(x) between the smallest ossible value of (often - ) and oint x.

11 STT315, Section 701, Summer 006 (1) Uniform Distribution: Density Function of Uniform Distribution: ~ U(a, b) if the density function of is 1/( b a) f ( x) = 0 Uniform Distribution Formulas: If ~ U(a, b), then P ( x1 1 a a x b all other x ) = ( x x ) /( b ), a x x. E ( ) = ( a + b / ), V ( ) = ( b a) / 1. x 1 b Examle: Suose ~ U(10,1), what are 10 11), >10.5)? () Exonential Distribution: Density Function of Exonential Distribution: ~ E(l) if the density function of is λe f ( x) = 0 λx x 0 x < 0 Exonential Distribution Formulas: If ~ E(l), then P ( x1 λx1 λx x ) = e e, x <. 0 1 x λx x) = 1 e. E ( ) = 1/ λ, V ( ) = 1/ λ. Examle: Suose ~ E(1.), what are 1 ), >0.5)? Examle: A articular brand of handheld comuters fails following an exonential distribution with a m of 54.8 months. The comany gives a warranty for 6 months. a. What ercentage of the comuters will fail within the warranty eriod? b. If the manufacturer wants only 8% of the comuters to fail during the warranty eriod, what should be the average life?