AI5002-Assignment 1 Probability and Random Variables Solved

I.      Problem

Find the MGF for X ˜N(µ,σ2) .

II.     Solution

 The Moment generating function for a Normal distribution is given as MX(t) = E[e−tX]. This is given by the Laplace transform Lx(t) of the density function f  Z ∞

1

Z ∞

−∞

Z ∞

t

                                            t22σ2 −tµZ ∞µ)2 dx                        (7)

= e

−∞ 2πσ2

Let y = x+ tσ2, dy = dx Integral under normal density=1. So,

Now,

                                    t2σ2 −tµZ ∞            1              µ)2

                                         Lx(t) = e 2dy                            (8)

−∞ 2πσ2

                                                                       = e−µt+σ22t2                     (9)

For the obtained expression, the MGF (0) =1. The same result is also obtained using the python code.

Download python code from here