Answered: Suppose that a random sample of size 20…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Suppose that a random sample of size 20 is taken from a normal distribution with unknown mean and known variance equal to 1, and the mean is found to be x ̄ = 10. A normal distribution was used as the prior for the mean, and it was found that the posterior mean was 15 and the posterior standard deviation was 0.1. What were the mean and standard deviation of the prior?

Bayesian Stats

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

Expert Solution

Step 1

Given Information:

Sample size $n = 20$

Variance $σ^{2} = 1$

Sample mean $\bar{x} = 10$

Posterior mean $μ_{p o s t} = 15$

Posterior standard deviation $σ_{p o s t} = 0.1$

To find the mean and standard deviation of the prior:

Standard deviation of the prior is obtained as follows:

Formula:

$\begin{array}{rcl} {σ^{2}}_{p o s t} & = & \frac{σ^{2} ({σ_{0}}^{2} / n)}{{σ_{0}}^{2} + (σ^{2} / n)} \\ 0 . 1^{2} & = & \frac{1 ({σ_{0}}^{2} / 20)}{{σ_{0}}^{2} + (1 / 20)} \\ 0.01 & = & \frac{{σ_{0}}^{2} / 20}{{σ_{0}}^{2} + 0.05} \\ 0.01 {σ_{0}}^{2} + 0.0005 & = & \frac{{σ_{0}}^{2}}{20} \\ {σ_{0}}^{2} & = & 0.2 {σ_{0}}^{2} + 0.01 \\ {σ_{0}}^{2} - 0.2 {σ_{0}}^{2} & = & 0.01 \\ 0.8 {σ_{0}}^{2} & = & 0.01 \\ {σ_{0}}^{2} & = & \frac{0.01}{0.8} \\ {σ_{0}}^{2} & = & 0.0125 \end{array}$

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