# Segment 22 Sanmit Narvekar

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## Segment 22

#### To Compute

1. In lecture slide 3, suppose (for some perverse reason) we were interested in a quantity $\displaystyle f = b_3/b_5$ instead of $\displaystyle f = b_3b_5$ . Calculate a numerical estimate of this new $\displaystyle f$ and its standard error.

The mean is simply the function evaluated at the fitted parameters:

$\displaystyle \langle f \rangle = \frac{b_3}{b_5} = \frac{0.6582}{1.4832} = 0.4438$

The variance can be calculated using the following formula:

$\displaystyle \text{Var}(f) = \nabla f \Sigma \nabla f^T$

The gradient of f is:

$\displaystyle \nabla f = \left[0, 0, \frac{1}{b_5}, 0, - \frac{b_3}{b_5^2} \right]$

By substituting this into the formula above and using the covariance matrix in the slides, we get the variance of f to be:

$\displaystyle \text{Var}(f) = \frac{1}{b_5^2}\Sigma_{33} + 2 \frac{1}{b_5} \left(- \frac{b_3}{b_5^2} \right) \Sigma_{35} + \frac{b_3^2}{b_5^4} \Sigma_{55}$

Plugging in the numbers and crunching gives a variance of 0.0145 and a standard deviation of 0.12044.

2. Same set up, but plot a histogram of the distribution of $\displaystyle f$ by sampling from its posterior distribution (using Python, MATLAB, or any other platform).

Here is the MATLAB code:


% Mean and cov of b's
bfit = [1.1235 1.5210 0.6582 3.2654 1.4832];

covar = [0.1349 0.2224 0.0068 -0.0309 0.0135;
0.2224 0.6918 0.0052 -0.1598 0.1585;
0.0068 0.0052 0.0049 0.0016 -0.0094;
-0.0309 -0.1598 0.0016 0.0746 -0.0444;
0.0135 0.1585 -0.0094 -0.0444 0.0948];

% Sample lots of b's
bs = mvnrnd(bfit, covar, 10000);

% Calculate f for each of those b's
fb = bs(:,3) ./ bs(:,5);

% Plot a histogram
hist(fb, 30)

% Report empirical mean and std dev
mean(fb)
std(fb)



This resulted in the following mean and standard deviation:


mu =

0.4720

stdDev =

0.1492



And here is the corresponding histogram of the posterior distribution of the parameter:

#### To Think About

1. Lecture slide 2 asserts that a function of normally distributed RVs is not, in general, normal. Consider the product of two independent normals. Is it normal? No! But isn't the product of two normal distribution functions (Gaussians) itself Gaussian? So, what is going on?

2. Can you invent a function of a single normal N(0,1) random variable whose distribution has two separate peaks (maxima)? How about three? How about ten?