# User:Tameem/Segment (9)

### Problems

#### To Compute

1. Use characteristic functions to show that the sum of two independent Gaussian random variables is itself a Gaussian random variable. What is its mean and variance?

Doing it the hard way (using Mathematica):

Let X and Y be two independent Gaussian random variables, X has a mean ${\mu}_1$, and standard deviation ${\sigma}_1$, and Y has a mean ${\mu}_2$, and standard deviation ${\sigma}_2$. Let S = X + Y.

$p_S(s) = \int_{-\infty}^{\infty} p_X(u)p_Y(s-u) du = -\frac{1}{2\pi{\sigma}_1{\sigma}_2}\int_{-\infty}^{\infty} e^{\frac{1}{2}((\frac{u-{\mu}_1}{{\sigma}_1})^2-(\frac{s-u-{\mu}_2}{{\sigma}_2})^2)} du$

Using Mathematica:

Clearly, it is a Guassian Distribution with mean = ${\mu}_1+{\mu}_2$, and variance ${{\sigma}_1}^2+{{\sigma}_2}^2$

2. Calculate (don't just look up) the characteristic function of the Exponential distribution.

$\Phi_X(t)= \int_0^{\infty} \beta e^{-(\beta-it)x} dx = \frac{\beta}{-(\beta-it)} e^{-(\beta-it)x}|_0^\infty = \frac{\beta}{-(\beta-it)}[0-1] = \frac{\beta}{(\beta-it)}$

#### Class Activity

Team 2, Teamed with with User:Noah, User:Kai, Keerthana Kumar's Solution, User:Jzhang, User:Trettels.

You can refer to: Kai for class activities.