27 let x POI Mu a Find the factorial moment generating funct

27. let x POI (Mu) (a) Find the factorial moment generating function (FMGF) of X, Gx(t). (b) Use Gx(t) to find E(X). (c) Use Gx(t) to find E[X(x - 1)). Suppose the X POI(10). (a) Find P[5

Solution

This is Poisson Distribution.

Mean = 10

For Poisson , = Var(X) = 10 ,

Therefore,

Standard Deviation = sqrt(10) = 3.1623

In general,

P(x; ) = (e^-) (^x) / x!

a)

P[5<X<15] = P[X<15] - P [X<=5]

= (e^-10) (10⁶) / 6! + (e^-10) (10⁷) / 7! + ........ + (e^-10) (10¹⁴) / 14!

= 0.9165 - 0.0671

= 0.8494 Answer

b)

P(5 < Y < 15) = p(10 5 < Y < 10 + 5)

= P(|Y 10| < 5) 1 sqrt(10)/25

= sqrt(10)/25

= 0.1265 Answer

c)

P[1-k< X/ < 1+k]

= P[-k< X < +k]

= P[|X-| < k] >= 1 - sqrt() / ^2k^2

=> P[|X-| < k] >= 1 - 1/ ^3/2k^2