STAT 400 Discussion 5 Spring 2015 1. Suppose that the proportion of genetically modified (GMO) corn in a large shipment is 2%. Suppose 50 kernels are randomly and independently selected for testing. a) Find the probability that exactly 2 of these 50 kernels are GMO corn. b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels are GMO corn. 2. Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 87, f ( k ) = k 31, k = 2, 4, 6, 8, … . ( possible values of X are even non-negative integers: 0, 2, 4, 6, 8, … ). Recall Discussion #1 Problem 2 (a): this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist? b) Find E ( X ). 3. Suppose a discrete random variable X has the following probability distribution: f ( 1 ) = ln 3 – 1, f ( k ) = ( )! 3lnkk, k = 2, 3, 4, … . ( possible values of X are positive integers: 1, 2, 3, 4, … ). Recall Discussion #1 Problem 2 (b): this is a valid probability distribution. a) Find µ X = E ( X ) by finding the sum of the infinite series. b) Find the moment-generating function of X, M X ( t ). c) Use M X ( t ) to find µ X = E ( X ). “Hint”: The answers for (a) and (c) should be the same.4. Suppose a discrete random variable X has the following probability distribution: f ( k ) = k 5100, k = 3, 4, 5, 6, … . Recall Discussion #1 Problem 3: this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist? b) Find E ( X ). 5. Let X be a continuous random variable with the probability density function f ( x ) = 3122 x, 4 ≤ x ≤ 10, zero otherwise. a) Find the probability P ( X > 9 ). b) Find the mean of the probability distribution of X. c) Find the median of the probability distribution of X. 6. Suppose a random variable X has the following probability density function: f ( x ) = 1 – x/ 2 , 0 ≤ x ≤ 2, zero elsewhere a) Find the cumulative distribution function F ( x ) = P ( X ≤ x ). b) Find the median of the probability distribution of X. c) Find the probability P( 0.8 ≤ X ≤ 1.8 ). d) Find µX = E ( X ). e) Find σX2 = Var( X ). f) Find the moment-generating function of X.1. Suppose that the proportion of genetically modified (GMO) corn in a large shipment is 2%. Suppose 50 kernels are randomly and independently selected for testing. a) Find the probability that exactly 2 of these 50 kernels are GMO corn. Let X = number of GMO kernels in a sample of 50. Then X has Binomial distribution, n = 50, p = 0.02. P( X = 2 ) = ( ) ( )482 0201020250..C −⋅⋅ ≈ 0.1858. b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels are GMO corn. Poisson Approximation to Binomial Distribution: λ = n ⋅ p = 50 ⋅ 0.02 = 1.0. P( X = 2 ) = ! 201012 .e.−⋅ ≈ 0.1839.2. Suppose a discrete random variable X has the following probability distribution: f ( 0 ) = 87, f ( k ) = k 31, k = 2, 4, 6, 8, … . ( possible values of X are even non-negative integers: 0, 2, 4, 6, 8, … ). Recall Discussion #1 Problem 2 (a): this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist? M X ( t ) = E ( e t X ) = ∑∞=⋅⋅ +12 2 0 31 87kktktee = ∑∞=+1 2 9 87kkte = 91987 2 2 ttee−+ = ttee 2 2 987−+ = 8199 2 −−te. Must have 9 2 te < 1 for geometric series to converge. ⇒ t < ln 3. b) Find E ( X ). M 'X ( t ) = ( ) ()( )2 2 2 2 2 2 9 29 2ttttteeeee−−−− = ()2 2 2 9 18ttee−, t < ln 3. E ( X ) = M 'X ( 0 ) = 6418 = 329. ORE ( X ) = ∑⋅x xpxall)( = 8642 38363432870 ++++⋅ + … 91 E ( X ) = 864 363432++ + … ⇒ 98 E ( X ) = 8642 32323232+++ + … = 91192− = 41. ⇒ E ( X ) = 329. OR E ( X ) = ∑⋅x xpxall)( = ∑∞=⋅⋅ +12 312 870 kkk = ∑∞=⋅⋅1 91 2kkk = ∑∞=−⋅⋅⋅119891 82 kkk = ( ) YE82⋅, where Y has a Geometric distribution with probability of “success” p = 98. ⇒ E ( X ) = ( ) YE82⋅ = 8982⋅ = 329.3. Suppose a discrete random variable X has the following probability distribution: f ( 1 ) = ln 3 – 1, f ( k ) = ( )! 3lnkk, k = 2, 3, 4, … . ( possible values of X are positive integers: 1, 2, 3, 4, … ). Recall Discussion #1 Problem 2 (b): this is a valid probability distribution. “Hint”: Recall that akkeka 0! =∑∞=. a) Find µ X = E ( X ) by finding the sum of the infinite series. E ( X ) = ∑⋅x xpx all)( = 1 ⋅ ( ln 3 – 1 ) + ()∑∞=⋅2 ! 3lnkkkk = ln 3 – 1 + ()( )∑∞−=2! 13lnkkk = ln 3 – 1 + ()()( )∑∞−=−⋅21! 13 3lnlnkkk = ln 3 – 1 + ()()∑∞=⋅1! 3 3lnlnkkk = ln 3 – 1 + ( )( ) 3 1 3ln ln −⋅e = 3 ln 3 – 1 ≈ 2.2958. b) Find the moment-generating function of X, M X ( t ). M X ( t ) = ∑⋅xxt xp eall)( = e t ⋅ ( ln 3 – 1 ) + ( )∑∞=⋅2 ! 3lnkkktke = e t ⋅ ( ln 3 – 1 ) + ∑∞=2 ! 3lnkktke = e t ln 3 – e t + 3ln tee – 1 – e t ln 3 = 1 3 −−tete. c) Use M X ( t ) to find µ X = E ( X ). “Hint”: The answers for (a) and (c) should be the same. ( )tteetet 3 3 M ln ' X−⋅⋅=, E ( X ) = () 0M' X = 3 ln 3 – 1.4. Suppose a discrete random variable X has the following probability distribution: f ( k ) = k 5100, k = 3, 4, 5, 6, … . Recall Discussion #1 Problem 3: this is a valid probability distribution. a) Find the moment-generating function of X, M …
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