Help on Probability - Density & Moment Generating Functions

[torres1234]

Help on Probability!@@ :)

Homework Statement

If we assume that X is a standard normal random variable with density function

f(x) = 1 / (2pi)^(1/2) * exp( - (x^2) / 2 )

what would be density functions of random variables of A = X^2 and B = exp (X)
and what would be the moment generating function of random variables X, A, and B

Homework Equations

∫_0^∞▒〖x^(α-1) e^(-x) dx〗 =∫_0^∞▒〖β^α x^(α-1) e^(-βα) dx〗

The Attempt at a Solution

 

[LightHero]

The density function of A = X^2 is fA(a) = 1/2 * (1/2π)^(1/2) * a^(-1/2) * e^(-a/2) The density function of B = exp(X) is fB(b) = (1/2π)^(1/2) * b^(-1/2) * e^-((ln(b))^2 / 2) The moment generating function of random variables X, A, and B are MX(t) = e^(t^2/2) MA(t) = e^(2t^2) MB(t) = e^( t + (t^2/2) )