Moment generating function The moment generating function of random variable X is given by (t ) E[etX ] Moment generating function The moment generating function of random variable X is given by (t ) E[etX ] (t ) etx p( x) if X is discrete x Moment generating function The moment generating function of random variable X is given by (t ) E[etX ] (t ) etx p( x) if X is discrete x (t ) etx f ( x)dx if X is continuous dE[etX ] d tX '(t ) E[ e ] E[ XetX ] dt dt '(0) E[ X ] dE[etX ] d tX '(t ) E[ e ] E[ XetX ] dt dt '(0) E[ X ] d (t ) E[ XetX ] E[ X 2 etX ] dt (2) (0) E[ X 2 ] (2) More generally, d k 1 tX (t ) E[ X e ] E[ X k etX ] dt ( k ) (0) E[ X k ] (k ) Example: X has the Poisson distribution with parameter l Example: X has the Poisson distribution with parameter l x l x l ( l ) e ( l ) e tX tx tx (t ) E[e ] x 0 e x 0 e x! x! t x t ( l e ) l l l et l ( ee 1) =e x 0 e e e x! Example: X has the Poisson distribution with parameter l x l x l ( l ) e ( l ) e tX tx tx (t ) E[e ] x 0 e x 0 e x! x! t x ( l e ) l l l et l ( et 1) =e x 0 e e e x! (t ) d [e l ( et 1) dt ] t l ( et 1) le e Example: X has the Poisson distribution with parameter l x l x l ( l ) e ( l ) e tX tx tx (t ) E[e ] x 0 e x 0 e x! x! t x ( l e ) l l l et l ( et 1) =e x 0 e e e x! (t ) d [e l ( et 1) dt ] l et el ( e 1) t ax a e x 0 x! t l ( et 1) '(0) l e e t 0 l '(0) l e e t l ( et 1) ''(0) l e e t l ( et 1) t 0 l le le e t t l ( et 1) t 0 l l2 t l ( et 1) '(0) l e e t l ( et 1) ''(0) l e e t 0 l t l ( et 1) le le e t Var ( X ) E[ X 2 ] E[ X ]2 l t 0 l l2 If X and Y are independent, then X Y (t ) E[et ( X Y ) ] E[etX etY ) ] E[etX ]E[etY ] = X (t )Y (t ) The moment generating function of the sum of two random variables is the product of the individual moment generating functions Let Y = X1+X2 where X1~Poisson(l1) and X2~Poisson(l2) and X1 and X1 are independent, then E[etY ] E[et ( X1 X 2 ) ] E[etX1 ]E[etX 2 ] =e l1 ( et 1) l2 ( et 1) e e ( et 1)( l1 l2 ) Let Y = X1+X2 where X1~Poisson(l1) and X2~Poisson(l2) and X1 and X1 are independent, then E[etY ] E[et ( X1 X 2 ) ] E[etX1 ]E[etX 2 ] =e l1 ( et 1) l2 ( et 1) e Y ~ Poisson(l1 l2 ) e ( et 1)( l1 l2 ) Note: The moment generating function uniquely determines the distribution. Markov’s inequality If X is a random variable that takes only nonnegative values, then for any a > 0, E[ X ] P( X a) . a Proof (in the case where X is continuous): E[ X ] xf ( x )dx a a xf ( x)dx xf ( x )dx xf ( x)dx a a a af ( x)dx a f ( x)dx aP ( X a ) Strong law of large numbers Let X1, X2, ..., Xn be a set of independent random variables having a common distribution, and let E[Xi] = m. then, with probability 1 X1 X1 ... X n m n as n . Central Limit Theorem Let X1, X2, ..., Xn be a set of independent random variables having a common distribution with mean m and variance s. Then the distribution of X 1 X 1 ... X n nm s n tends to the standard normal as n . That is a X 1 X 1 ... X n nm 1 x2 / 2 P( a) e dx s n 2 as n . Conditional probability and conditional expectations Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y=y is defined as P{ X x, Y y} p( x, y ) p X |Y ( x | y ) P{ X x | Y y} . P{Y y} p( y ) for all values of y for which P(Y=y)>0. Conditional probability and conditional expectations Let X and Y be two discrete random variables, then the conditional probability mass function of X given that Y=y is defined as P{ X x, Y y} p( x, y ) p X |Y ( x | y ) P{ X x | Y y} . P{Y y} p( y ) for all values of y for which P(Y=y)>0. The conditional expectation of X given that Y=y is defined as E[ X | Y y] xP{ X x | Y y} xpX |Y ( x | y). x x Let X and Y be two continuous random variables, then the conditional probability density function of X given that Y=y is defined as f ( x, y ) f X |Y ( x | y ) . fY ( y ) for all values of y for which fY(y)>0. The conditional expectation of X given that Y=y is defined as E[ X | Y y] xf X |Y ( x | y)dx. E[ X ] E[ E[ X | Y y ]] E[ E[ X | Y ]] E[ X ] E[ X | Y y ]P(Y y ) if Y is discrete y E[ X ] E[ X | Y y ] f ( y ) dy if Y is continuous. Proof: