# Eleisha's Segment 9: Characteristic Functions

To Calculate:

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?

The characteristic function of the sum of independent random variables is the product of their characteristic functions.

${\text{The characteristic function of S is:}}\phi _{S}(t)$ $X\sim Normal(\mu _{1},\sigma _{1})$ $Y\sim Normal(\mu _{2},\sigma _{2})$ So let S = X + Y (the sum of two independent Gaussian random variables)

$\phi _{S}(t)=\phi _{X}(t)\phi _{Y}(t){\text{ since X and Y are independent}}$ $\phi _{n}ormal(t)=e^{i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}}$ $\phi _{S}(t)=e^{i\mu _{1}t-{\frac {1}{2}}\sigma _{1}^{2}t^{2}}\cdot e^{i\mu _{2}t-{\frac {1}{2}}\sigma _{2}^{2}t^{2}}$ 2. Calculate (don't just look up) the characteristic function of the Exponential distribution.

$\phi _{X}(t)=\int _{0}^{\infty }e^{itx}\cdot \lambda e^{\lambda x}dx$ $\phi _{X}(t)=\lambda \int _{0}^{\infty }e^{-(\lambda -it)x}dx$ $\phi _{X}(t)=\lambda \lim _{u\to \infty }\left[\int _{0}^{u}e^{-(\lambda -it)x}dx\right]$ $\phi _{X}(t)=\lambda \lim _{u\to \infty }\left[{\frac {-1}{\lambda -it}}\right]e^{-(\lambda -it)x}{\Big |}_{0}^{u}$ $\phi _{X}(t)=\lambda \left[\lim _{u\to \infty }e^{-(\lambda -it)u}+{\frac {1}{\lambda -it}}\right]$ $\phi _{X}(t)=\lambda \left[{\frac {1}{\lambda -it}}\right]$ $\phi _{X}(t)={\frac {\lambda }{\lambda -it}}$ 