Segment 9 Sanmit Narvekar

From Computational Statistics Course Wiki
Jump to navigation Jump to search

Segment 9

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?

From the slides, we know that the characteristic function of the sum of two independent random variables S = X + Y with distributions p_X(t) and p_Y(t) respectively is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_S(t) = \phi_X(t) \phi_Y(t)}

We also know the characteristic function of a Gaussian random variable is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_X(t) = e^{i \mu_X t - \frac{1}{2} \sigma_X^2t^2}}

So we plug in and simplify:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{X+Y}(t) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \phi_X(t) \phi_Y(t)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = e^{i \mu_X t - \frac{1}{2} \sigma_X^2t^2} e^{i \mu_Y t - \frac{1}{2} \sigma_Y^2t^2} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = e^{i t ( \mu_X + \mu_Y) - \frac{1}{2} t^2 (\sigma_X^2 + \sigma_Y^2)}}

By comparing the form of this equation to the characteristic function of a Gaussian random variable, its easy to see the result is another Gaussian random variable with mean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_X + \mu_Y} and variance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2_X + \sigma^2_Y} .


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

Since the exponential distribution's domain is the set of non-negative reals, we integrate from 0 to infinity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\infty e^{itx} \beta e^{-\beta x} dx }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \beta \int_0^\infty e^{x(it - \beta)} dx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\beta}{it - \beta} e^{x(it - \beta)} \Big|_0^\infty}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = - \frac{\beta}{it - \beta}}


To Think About

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)?

How would you do numerical procedures with complex numbers...?

Comments