Let Z N0 1 and show that Mzt EexptZ exp t22 t R Let X Nm

Let Z ~ N(0, 1) and show that M_z(t) = E[exp{tZ}] = exp {t^2/2}, t R. Let X ~ N(mu, sigma^2) and show that M_x(t) = E[exp{tX}] = exp {mu t + sigma^2/2 t^2}, t R.

Solution

b) A random variable X is said to be normal distribution having the parameters µ and ² then the probability density function is given by

f(x) = 1/ 2 {e^{-1/2((x-µ)/)2}}

=0otherwise

Where µ and ² are mean and variance of the normal distribution

The moment generating function is given by

= e^tx f(x) dx

= e^tx 1/ 2 {e^{-1/2((x-µ)/)2}} dx

Assume Z=((x-µ)/) and x=Z+µ

= e^tµ/2 e^{-1/2(z)^2-2tz)}} dz

Add and substrate t²² in the exponential function

= e^tµ/2 e^{-1/2(z)^2-2tz+t^2^2- t^2^2)}} dz

= e^tµ/2 e^{-1/2(z)^2-2tz+t^2^2- t^2^2)}} dz

= e^tµ/2 (e^{-1/2(z-t)^2}) e ^{-(t^2^2)}} dz

Let y=z - t

Since e^{-x^2/2} dx = 2

if XN(µ , ²) then standard normal variate is given by Z=((x-µ)/)

= e^-tµ/ e^{tµ+(t^2^2)/2}

Here ZN(0,1) hence taking µ = 0 and ² = 1 we get