STAT 400 1 Spring 2015 Discussion 5 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 7 8 f k 1 k 3 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 ln 3 k k 2 3 4 f 1 ln 3 1 f k k 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 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 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 100 k 5 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 x2 312 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 X E X f Find the moment generating function of X e Find X2 Var 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 50 C 2 0 02 2 1 0 02 48 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 n p 50 0 02 1 0 P X 2 1 0 2 e 1 0 0 1839 2 2 Suppose a discrete random variable X has the following probability distribution f 0 7 8 f k 1 k 3 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 a this is a valid probability distribution Find the moment generating function of X M X t For which values of t does it exist 7 e 2 t 7 1 M X t E e t X e 0 t e 2k t 8 k 1 9 8 k 1 3 2k e 2t 7 8 9 1 e 2t Must have 9 b e 2t k e 2t 7 9 1 8 9 e 2t 9 e 2t 8 9 1 for geometric series to converge t ln 3 Find E X M X t 2e 2t 9 e 2t e 2t 2e 2t 9 e 2t 2 18 9 E X M X 0 64 32 OR 18 e 2 t 9 e 2t 2 t ln 3 E X x p x all x 7 2 4 6 8 0 8 3 2 3 4 36 38 2 1 E X 9 3 4 4 3 6 6 3 8 2 9 2 2 2 2 8 1 E X 1 2 4 6 8 9 4 3 3 3 3 1 9 E X 9 32 OR E X x p x all x 2 1 k 8 k 1 9 7 1 1 0 2k 2 k 8 k 1 3 2k 9k k 1 k 1 2 8 E Y 8 9 where Y has a Geometric distribution with probability of success p E X 2 2 9 9 E Y 8 8 8 32 8 9 3 Suppose a discrete random variable X has the following probability distribution ln 3 k k 2 3 4 f 1 ln 3 1 f k k possible values of X are positive integers 1 2 3 4 Recall Discussion 1 Problem 2 b this is a valid probability distribution Hint a Recall that ak ea k 0 k Find X E X by finding the sum of the infinite series x p x 1 ln 3 1 E X all x ln 3 1 ln 3 k k 1 k ln 3 k k k 2 ln 3 1 ln 3 k 2 ln 3 1 ln 3 k 1 b 3 ln 3 1 k 1 k ln 3 1 ln 3 e ln 3 1 2 2958 Find the moment generating function of X M X t MX t e t x p x e t ln 3 1 Use M X t to find X E X M X t 3 et ln 3 e t e t k 2 all x t e ln 3 e t ln 3 1 k k 2 e t e t 1 3 c ln 3 k 1 k 2 ln 3 k etk ln 3 k k k e t ln 3 e t e e t ln 3 1 e t ln 3 Hint The answers for a and c should be the same E X M X 0 3 ln 3 1 4 Suppose a discrete random variable X has the following probability distribution f k 100 k 5 k 3 4 5 6 Recall Discussion 1 Problem 3 a this is a valid probability distribution Find the moment generating function of X M X t For which values of t does it exist MX t e t x p x all x 1 e t k 100 5 k 3 k et 100 k 3 5 k 3 et 100 5 first term 1 base et 1 5 Geometric series if b et 5 1 4 e 3t 5 et t ln 5 Find E X 12 e 3 t 5 e t 4 e 3 t e t 3t 4t 60 e 8 e M X …
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