Question

工、エ.D ARE CONTINUOUS RANDOM VARIABLES ACCORDIN IN TO 7IS OISTRI BUTION AND VAR TABLE DISCRETE A RANOOM PO SITLVE LET Y=X TERMS OF YIN MGF OF ERPRESS AND LJ

if Xn are iid continuous random variables in n according to the PDF of fx , and Z is a positive discrete random variable according to Y= sum of Xn. Find the MGF of Y in terms of Z and X

Answer 1

the m.g.f of xi is M_xi(t)=E(e^txi) , i =1 to z

now ,

$y=\sum_{i=1}^{z}xi$

where z is positive discrete random variable.

now , m.g.f of y is

E(y) = E(e^ty)

=E(e^{t(x₁+x₂+ ........+x_z)})

=E(e^tx₁  . e^tx₂ ......... e^tx_z) [as, x₁,x₂, ..... ,x_z are i.i.d. random variables]

=E(e^tx₁) . E(e^tx₂). .... E(e^tx_z)

$=\prod_{i=1}^{z}E(e^t^x^i)$

$=\prod_{i=1}^{z}Mxi(t)$ ...................... (Ans)