# Difference between revisions of "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.

$\displaystyle \text{The characteristic function of S is:} \phi_S(t)$

$\displaystyle X \sim \text{Normal}(\mu_1, \sigma_1)$

$\displaystyle Y \sim \text{Normal}(\mu_2, \sigma_2)$

So let S = X + Y (the sum of two independent Gaussian random variables)

$\displaystyle \phi_S(t) = \phi_X(t) \phi_Y(t) \text{ since X and Y are independent}$

$\displaystyle \phi_{X}(t) = e^{i \mu_1 t - \frac{1}{2} \sigma_1^2 t^2 }$

$\displaystyle \phi_{Y}(t) = e^{i \mu_2 t - \frac{1}{2} \sigma_2^2 t^2 }$

$\displaystyle \phi_S(t) = e^{i \mu_1 t - \frac{1}{2} \sigma_1^2 t^2 } \cdot e^{i \mu_2 t - \frac{1}{2} \sigma_2^2 t^2 }$

$\displaystyle \phi_S(t) = e^{i (\mu_1 + \mu_2) t - \frac{1}{2} (\sigma_1^2 + \sigma_2^2) t^2 }$

$\displaystyle \text{Notice that this characteristic function is just the characteristic function of a Normal with mean } (\mu_1 + \mu_2) \text{ and variance } (\sigma_1^2 + \sigma_2^2)$

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

$\displaystyle \phi_X(t) = \int_0^\infty e^{itx} \cdot \lambda e^{\lambda x} dx$

$\displaystyle \phi_X(t) = \lambda \int_0^\infty e^{-(\lambda - it)x} dx$

$\displaystyle \phi_X(t) = \lambda\lim_{u \to \infty} \left[ \int_0^u e^{-(\lambda - it)x} dx \right]$

$\displaystyle \phi_X(t) = \lambda \lim_{u \to \infty} \left[ \frac{-1}{\lambda - it}\right] e^{-(\lambda - it)x}\Big|_0^u$

$\displaystyle \phi_X(t) = \lambda \left[ \lim_{u \to \infty} e^{-(\lambda - it)u} + \frac{1}{\lambda - it} \right]$

$\displaystyle \phi_X(t) = \lambda \left[ \frac{1}{\lambda - it}\right]$

$\displaystyle \phi_X(t) = \frac{\lambda}{\lambda - it}$

1. Learn enough about contour integration to be able to make sense of Saul's explanation at the bottom of slide 7. Then draw a picture of the contours, label the pole(s), and show how you calculate their residues.

2. Do you think that characteristic functions are ever useful computationally (that is, not just analytically to prove theorems)?

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