STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Example 2 3 Bonus Let X be a discrete random variable with p m f p x that is positive on the odd non negative integers 1 3 5 7 9 and is zero elsewhere Suppose p 1 c unknown a p k 1 k 2 k 3 5 7 9 Find the value of c that makes this is a valid probability distribution p x 1 Must have all x 1 p 1 p 3 p 5 p 7 p 9 c b 1 23 1 25 1 27 1 29 1 8 c 1 1 4 c 1 6 5 6 c Find E X E X x p x all x 5 3 5 7 9 6 23 25 27 29 5 3 5 7 24 2 5 2 7 2 9 1 E X 4 2 5 3 5 2 2 2 13 3 E X 1 32 1 6 8 24 2 5 2 7 2 9 12 4 1 4 E X 13 9 OR MX t E e t X e 1t 5 2 k 1 t 1 e 2 k 1 6 k 1 2 e 2t k 5 e t e 2 t 5 et e t e t 4 2t 6 2 k 1 4 6 2 e 1 4 M X t 5 t et e 2t 5 e 3t e e t 6 2 4 e 2t 6 8 2e 2t 5 t 3e 3t 8 2 e 2 t e 3t 4 e 2 t e 2 6 8 2e 2t t ln 2 5 t 24 e 3 t 2 e 5 t e 6 2t 2 8 2e t ln 2 E X M X 0 5 22 52 13 6 36 36 9 OR MX t E e t X et 1t 5 e 6 5 t e 6 2 k 1 4 1 t 4et e 3 8 2e 2t 1 t 5 t e 1 e 1 2 t 6 2 1 e 4 2t k e M X t e 2 k 1 t 2 k 1 2 k 1 t ln 2 1 t 4e t 8 2e 2t 4et 4e 2t e 2 3 8 2e 2t 1 t 32 e t 8 e 3 t e 3 2t 2 8 2e t ln 2 1 40 52 13 E X M X 0 3 36 36 9 2 Let X be a discrete random variable with p m f k 3 p k c 4 a k 5 6 7 8 Find the value of c that makes this is a valid probability distribution Must have p x 1 k 5 4 all x 5 3 3 k first term 4 4 1 base 3 k 5 1 4 k 3 k 3 c c 1 243 1024 1 4 k 5 4 243 256 OR 3 4 k 5 k 3 k 0 4 1 1 b 3 4 k 3 9 27 81 1 4 16 64 256 256 192 144 108 81 781 1024 781 243 4 256 256 256 256 256 256 256 256 44 256 c 243 35 Find P X is even P X is even p 6 p 8 p 10 p 12 6 8 3 3 3 c c c 4 4 4 first term 1 base 10 3 c 4 6 3 c 3 16 2 7 3 1 16 4 4 12 3 0 42857 7 OR 6 8 3 3 3 P X is even c c c 4 4 4 5 7 10 9 3 c 4 3 3 3 3 P X is odd c c c c 4 4 4 4 c 3 P X is odd 4 P X is even 1 P X is odd P X is even P X is even 12 11 4 P X is even 3 P X is odd 7 P X is even 3 3 0 42857 7 Find the moment generating function of X M X t For which values of t does it exist 256 3 e k t 243 4 k 5 MX t E e t X k 256 3 e t 243 k 5 4 k 5 3e t 4 256 243 3e t 1 4 d 256 243 e 5 t e 5t 243 1024 768 e t 4 3e t t ln 4 3 t ln 4 3 Find E X M X t 5e 5 t 4 3e t e 5 t 3e t 4 3e t 2 20 e 5 t 12 e 6 t 4 3e t 2 E X M X 0 8 OR E X x p x 5 all x 5 5 8 6 7 8 5 6 7 1 256 3 256 3 256 3 256 3 256 3 E X 4 4 243 4 243 4 243 4 243 4 243 4 1 7 256 3 256 3 256 3 5 6 7 243 4 243 4 243 4 3 E X 4 6 256 3 256 3 256 3 256 3 6 7 8 243 4 243 4 243 4 243 4 k 256 3 1 1 2 243 k 5 4 E X 8 OR E X x p x all x k 5 k 256 3 243 4 k 256 1 3 3 k 243 4 4 k 5 k 1 k 1 2 3 256 1 3 1 1 3 1 3 1 3 k 1 2 3 4 81 k 1 4 4 4 4 4 4 4 4 4 256 1 6 27 27 256 94 E Y E Y 81 4 16 64 64 81 64 where Y has a Geometric distribution with probability of success p E X 1 4 256 94 256 94 256 162 8 E Y 4 81 64 81 64 81 64 8 OR k 3 p k c k 5 6 7 8 4 c 44 3 k 5 1 3 p k X Geometric p 4 k 5 6 7 8 4 4 5 1 4 1 Let Y have a Geometric p distribution 4 X Y 4 c 1 t e et 4 MY t t 3 t 4 3 e 1 e 4 Theorem t ln 3 4 ln 4 3 Then M V t Let V a W b e b t M W a t MV t E e t V E e t a W b E e a t W e b t Proof eb t E ea t W MX t MY 4 t e b t M W a t e4 t MY t e 4t et 4 3e t e 5t 4 3e t t ln d E X E Y 4 4 4 8 4 3 3 Suppose a discrete random variable X has the following probability distribution P X k a ln 2 k k k 1 2 3 Verify that this is a valid probability distribution x p x 0 p x 1 all x ln …
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