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Unformatted text preview: Chapter 4 Further Topics on Random Variables 1 Transforms The moment generating function (MGF) of a random variable X is M X ( t ) = E [ e tX ] . In the discrete case, M X ( t ) = X x e tx p X ( x ) . In the continuous case, M X ( t ) = Z  e tx f X ( x ) dx. Note that in general M X ( t ) may not be defined for all values of t , however M X (0) is always defined and equals 1. Moment generating functions for some common random variables Bernoulli p : p X ( x ) = p x (1 p ) 1 x , x = 0 , 1 M X ( t ) = 1 p + pe t Binomial ( n, p ): p X ( x ) = n x p x (1 p ) n x , x = 0 , 1 , . . . , n M X ( t ) = (1 p + pe t ) n 1 2 MTH2222 Mathematics of Uncertainty Discrete Uniform over [ m, n ]: p X ( x ) = 1 n m + 1 , x = m, . . . , n M X ( t ) = e ( n +1) t e mt ( n m + 1)( e t 1) Geometric p : p X ( x ) = p (1 p ) x 1 , x = 1 , 2 , . . . M X ( t ) = pe t 1 (1 p ) e t , t < ln(1 p ) Poisson : p X ( x ) = e x x ! , x = 0 , 1 , . . . M X ( t ) = e ( e t 1) Uniform over [ a, b ]: f X ( x ) = 1 b a , a x b M X ( t ) = e bt e at ( b a ) t Chapter 4 3 Exponential : f X ( x ) = e x , x > M X ( t ) =  t , t < Gamma ( , ): f X ( x ) = ( ) x  1 e x , x > M X ( t ) =  t , t < Normal ( , 2 ): f X ( x ) = 1 2 exp ( x ) 2 2 2 , x > M X ( t ) = exp ( t + 2 t 2 / 2 ) If M X ( t ) = M Y ( t ) < + for all values of t in an open interval containing 0, then X and Y have the same CDF (distribution). In other words, if M X ( t ) is finite for all values of t in an open interval containing 0, then M X ( t ) determines uniquely the CDf (distribution) of X . Identify the distributions of the following random variables: * M U ( t ) = 2 3 + 1 3 e t 4 * M V ( t ) = exp ( t 2 2 t ) * M W ( t ) = e t 2 e t 4 MTH2222 Mathematics of Uncertainty * M X ( t ) = 1 1 2 t * M Y ( t ) = sinh t t * M Z ( t ) = e t 2 3 + 1 3 e t 4 If the MGF of X can be expressed as M X ( t ) = p 1 e tx 1 + p 2 e tx 2 + . . . + p n e tx n = n X k =1 p k e tx k , then X has a discrete distribution with PMF P [ X = x k ] = p k , k = 1 , . . . , n. M X ( t ) = 2 3 + 1 3 e t 2 M aX + b ( t ) = e bt M X ( at ). If Z is standard normal and X = Z + , then X is normal ( , 2 )....
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This note was uploaded on 10/09/2010 for the course MTH 2222 taught by Professor Kaizhamza during the Two '10 term at Monash.
 Two '10
 KaizHamza

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