Question

Give one example of solving a problem using mgf of normal distribution

Answer 1

A continuous random variable X is said to have a Normal Distribution if its PDF is:

$f_X(x)=\frac{1}{σ \sqrt{2π}}e^{−\frac{1}{2}(\frac{x−μ}{σ})^2},x∈R\\$

So if X∼N(μ,σ²) then it’s MGF is:

$M_x(t)\\ =E(e^{tx})\\ =∫^∞_{−∞}e^{tx}\frac{1}{σ \sqrt{2π}}e^{−\frac{1}{2}}(\frac{x−μ}{σ})^2dx\\ =∫^∞_{−∞}e^t(μ+σz)\frac{1}{\sqrt{2π}}e−\frac{1}{2}z^2dz, where \frac{x−μ}{σ}=z\\ =(e^{tμ}+\frac{1}{2}t^2σ^2)∫^∞_{−∞}\frac{1}{\sqrt{2π}}e^−\frac{1}{2}(z−tσ)^2dz,$

Note that the portion under integration sign represents PDF of N(tσ,1) distribution; so eventually the value of the integration will be 1

$e^{tμ}+\frac{1}{2}t^2σ^2, t ∈R$

So, $M_x(t)= e^{tμ}+\frac{1}{2}t^2σ^2, t ∈R$