Transform Techniques A convenien way of finding he momens of a random variable is he momen generaing funcion (MGF). Oher ransform echniques are characerisic funcion, z-ransform, and Laplace ransform. Momen Generaing Funcion For a real, he MGF of he random variable is M E[ e ] e Example - Bernoulli k x k e p ( x ) discree x k e f ( x) dx coninuous The probabiliy mass funcion is p p, x ( x) p, x M e e ( p) + e p pe + p is a real variable Example - Exponenial M e e f ( x) dx x x x e e dx ( ) x ( ) e dx <
Properies of MGF ) Find momens easily from he MGF Recall 3 3 e + + + +! 3! Taking expecaion, 3 3 M e + + + +! 3! Differeniaing wih respec o, m imes, m ( m) d m M () M ( ) he m - h momen m d, ) Can show wo random variables have he same probabiliy disribuion M M f( u) f( u) If wo random variables and have he same MGF, and have he same probabiliy disribuion. 3) Convergence Consider a sequence of random variables,, wih cdf F ( x), F ( x), and heir momen generaing funcions M ( ), M ( ),. F ( x) F( x) iff M ( ) M( ) n n.
3 4) Sum of Independen RVs Le and be independen random variables wih M ( ) and M ( ) respecively. Define Z +. Proof: M M M Z M e Z Z e ( + ) e e if and are independen, e M M e 5) Ohers If a + b, hen b M e M( a).
4 Bernoulli RV p p for x ( x) p for x M e pe + pw M (), which mus be rue for any. () () M pe M () p M pe M () p In fac, M pe + p 3 p+ p + + + +! 3! 3 + p + p + p +! 3! Recall for any random variable, Therefore 3 3 M + + + +! 3! n p for n,,
5 Poisson RV k e Le be a Poisson random variable wih [ ] P k for k,,,. k! The MGF is M e ( e ) Homework. Derive he MGF of he Poisson rv. Propery - Sum of independen Poisson is Poisson When and are independen Poisson wih arrival raes and respecively, heir MGF are Thus ( e ) ( e ) and. M e M e M M M + e ( + )( e ) Eq. p shows is Poisson wih rae +. + ( p) Noe. Addiion of independen Poisson packe sreams yields a Poisson packe sream.
6 Exponenial RV ( ) Le ~ Exp. The MGF is M < Noe M 3 + + + + 3 3! 3! + + + +!! 3 3! n n! n Propery Sum of Independen Exponenial Wha is he sum of independen exponenial? Suppose M, M. M + + is no exponenial, bu becomes -Erlang when : M +
7 Propery Exponenial and Gamma Funcion Relaion beween he momens of an exponenial random variable and he gamma funcion. Le be an exponenial random variable wih. we know n which means Eq, e shows n! n n x ( ) x e dx n! e Γ ( n+ ) x e dx n! n x
8 Gaussian RV ( σ ) Le ~ N m,. he MGF is M e σ m+ Homework. Derive he MGF of he Gaussian rv. Propery - Gaussian Sum of independen Gaussian is Gaussian.* Proof. ( ) e σ σ m+ m + Le M ( ) e and M ( ) e. If and are independen, M M M. + ( σ + σ ) m + m + ( σ σ ) Eq. g shows + is Gaussian, N m + m, +. ( g) Also noe ha ( ) M e ( σ + σ ) m m + Noe In fac, any linear combinaion of joinly Gaussian random variables is Gaussian. Suppose and are joinly Gaussian random variables. For any consans a, b and c, define V a + b + c V is a Gaussian random variable. and do no have o be independen for V o be Gaussian.
9 Gamma RV Le ~ Gamma( α, ) for any > and α>, wih is pdf α ( x) x f ( x) e for x. Γ( α) The gamma funcon is defined as z x Γ ( z+ ) x e dx he MGF is Proof. M e M for <. ( ) α α α x ( x) x e e dx Γ( α) α α ( ) x (( ) x) e d x Γ( α) (( ) ) provided > α Γ( α) ( ) α α α for <. y y e dy
Propery Relaion beween Gamma and Erlang Disribuions ( ) m-erlang is a special case of Gamma α, where α is a posiive ineger m. Le m be a posiive ineger, and assume ha,,, are independen exponenial random variables wih arrival rae. m Define + + + m. is he m-h arrival ime in a Poisson arrival process wih arrival rae. is referred o as an m-erlang random variable. { j} ( ) Since is Exp, Since. j are independen, M. The MGF of is idenical o he MGF of Gamma, wih α m. ( α) Gamma( α) An m-erlang random variable is a special case of, where j α M m is a posiive ineger m.
Propery Sum of independen gamma is gamma. ( αx) ( αy) Le ~ Gamma, and ~ Gamma,. Assume and are independen. ( αx αy) + ~ Gamma, + Proof: αx αy M, M αx+ αy M+ ( αx αy) + ~ Gamma, + Example. m-erlang Suppose α and α are posiive inegers. x y For example, suppose α 3 and α 4. x is he ime when he 3rd arrival occurs, is he ime when he 4h arrival occurs, and + is he ime when he 7h arrival accurs. y
Chi-square wih degree of freedom Le ~ N(,) Define. ~ Chi. Chi - square disribuion of degree of freedom / / M <. / Homework. MGF of Chi Derive he MGF of Chi(). Propery. Chi is a special case of gamma. α M Special case of Gamma, α Find pdf of : Since Chi() Gamma,, α ( y) f ( y) e Γ( α ) y y e y > π y and α
3 Chi-square wih k degrees of freedom k ( ) k / ( ) Le,, be independen N,. Define + + +. k ~ Chi k, Chi - square disribuion wih k degrees of freedom. / k M <. special case of Gamma, α / Proof. Since j is N (, ), j is Chi( ), and hus is Gamma,. { j} { j} Since are independen, are independen. Also sum of independen Gamma is Gamma. k Therefore is Gamma,. M / / k / <. Find pdf of : k Since Chi ( k ) Gamma,, α ( y) f ( y) e Γ( α) y k and α
4 Sum of a Random Number of Random Variables Le,, be iid random variables wih mean and σ. Define a new random variables as + + + where N is a random variable wih N and σ. N ( ) N σn VAR N σ + N Proof. Le M, M, M denoe MGF of,, N. N ( + + N e ) [ ] N is a posiive ineger random variale wih P N n, n,,. The MGF of is M e ( + + n P N n e ) n n n n [ ] [ ] P N n e [ ] [ ] n P N n M P N n e Define u log M. [ ] M P N n e n e u N N n ( ) log M n un ( ) ( ) M u e
5 ( e ) Differeniaing, ( ) M MN u u u MN u ( u) ( ) log M MN u M u M ( e) ( ) ( ) ( ) ( e ) Noe ha as, u u log M log. Therefore from, which saes ( ) M MN u M u M ( ) u N Wrie ( e ) as ( e ) Differeniaing 3 wih respec o, M u M M MN ( u) e ( 3) M M ( ) ( ) M MN u u MN u + M u u M u M Recalling, M N M M M M M M M M ( u) MN ( u ) + u M u M Subsiuing, and hus u, ( ) N + N N + Nσ
6 σ σ N + N N Nσ + N N Nσ + σn Example. Queue Lengh A queue conains N packes. Each packe conains bis. σn N σ N and are random variables. Assume N, σ, and are known. Le indicae he number of bis in he queue: + + N N VAR( ) N σ +. If σ, ha is, packes are of a fixed lengh, hen N, N and VAR( ) σ.