Section 3.3-3.5 Expected Value, Binomial, and Hypergeometric Random Variables Example 2. Poker hands. Hand Obtained Probability Payoff Let X be the payoff Nothing 0.501177 0 earned with one poker One Pair 0.422569 0.5 hand. What is the pmf of Two Pairs 0.047539 2 X? Three of a Kind 0.021128 5 Straight 0.003925 10 If you play the game Flush 0.001965 20 many times what average Full House 0.001441 25 payoff would you Four of a Kind 0.00024 100 expect? Straight Flush 0.000014 1000 Royal Flush 0.000002 10000 In lab, compared to payoff of 10, 100 games. The Expected Value of a Function Let X be a discrete random variable with p.m.f. p(x). The expected value or mean of X is given by E( X ) X x p ( x) x •If X is the payoff of a poker hand, what is E(X)? •If X is the sum of two six sided die, what is E(X)? Example 3. •Let X be the sum of two fair 6-sided dice. What are the possible values of X2? What is the probability of each of these outcomes? •Calculate E(X2). •Expected value of a function. Let X be a discrete r.v. with p.m.f. p(x) and let h(x) be a function. Then E (h( X )) h( x) p( x) x Mean, Variance and Standard Deviation • Let X be a discrete random variable. • We define the mean of X to be E[X] (usually denoted ). • We define the variance of X to be 2 = E[(X )2] = …=E[X2] E[X]2. (The first equality is taken as the definition of variance. The last formula is used for computations.) • We define the standard deviation of X to be = 2 . • These are theoretical versions of the sample mean, sample variance and sample standard deviation introduced for data. Example 3. Continued. Let X be the sum of two fair six sided die. Recall E(X). and E(X2). Calculate the variance from the definition. Calculate the variance from the shortcut formula. What is the standard deviation? In Minitab, generate 100 realizations of X. Calculate the mean and standard deviation of the data. Summary of 3.3 • Use the p.m.f. to calculate expected value of a random variable. • Calculate expected value of a function of a random variable. • Mean, Variance, and Standard Deviation of a random variable (, 2, ). • Mean, variance, standard deviation of a data set (x_bar, s2, s) Binomial Random Variables • Recall what binomial models. • Think of an application of the binomial r.v. • What is the pmf of a binomial r.v.? Note instead of p(x), pmf is labeled b(x;n,p). CDF of Binomial R.V. • Instead of F(x), cdf is labeled B(x;n,p). The cdf is tabulated in A.1. Part given here. • Suppose X is Bin(n,p), n=10 – find P(X2) from pmf. x p=.01 p=.05 p=.1 – find P(X2) from cdf. 0 1 0.904 0.996 0.599 0.914 0.349 0.736 – Find P(2 X 5) from pmf. 2 1.000 0.988 0.930 3 1.000 0.999 0.987 – Find P(2 X 5) from cdf. 4 1.000 1.000 0.998 5 1.000 1.000 1.000 6 1.000 1.000 1.000 7 1.000 1.000 1.000 8 1.000 1.000 1.000 9 1.000 1.000 1.000 10 1.000 1.000 1.000 Mean and Variance of Binomial • Give summations necessary for finding mean and variance of a random variable. • Summations simplify to – The mean of a binomial is np. – The variance is np(1-p). Hypergeometric Random Variables • Recall what hypergeometric probabilities are used for. Think of an application of hypergeometric probabilities. • Let X be the number of successes obtained when sampling from a population of size N with M successes and N-M failures. X is a hypergeometric r.v. What is the pmf of X? Note instead of p(x), pmf is labeled h(x;n,M,N). Mean and Variance • The mean of a hypergeometric r.v. is M n N • The variance is np(1-p). N n M M n 1 N 1 N N Relationship between binomial and hypergeometric. • One application of binomial is sampling from finite population of S and F with replacement. • While the hypergeometric is sampling w/o replacement. Recall HW problem: • 10000 boards, 2000 of which are green. Sample 2 boards . Are the events “1st board is green” and “second board is green” independent if sampling is done – w/ replacement – w/o replacement – Suppose instead there are only 10 boards, 2 of which are green and the sampling is done w/replacement. – W/o replacement Compare mean and variance. • Set p=M/N. Sample size of n. • What is the expected number of successes if sampling is done with replacement? Without replacement? • What is the variance of the number of successes if sampling is done with replacement? Without replacement? Notes • You should know and/or be able to derive pmfs of Binomial and Hypergeometric random variables. • You should know mean and variance of each. • You should be able to use tabulated cdf for Binomial random variables. • You are not responsible for the negative binomial r.v. described in section 3.5 or the Poisson r.v described in section 3.6. However, you should be able to work with these and potentially other random variables if details are given.
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