Consider n independent random variables with gaussian probability densities but witheach having its own mean t; and standard deviation o;.Find the probability density of the sum of those n variables as well as its mean andstandard deviation.

Consider n independent random variables with gaussian probability densities but with each having its own mean ; and standard deviation o;. Find the probability density of the sum of those n variables as well as its mean and standard deviation.

Consider n independent random variables with gaussian probability densities but with each having its own mean ; and standard deviation o;. Find the probability density of the sum of those n variables as well as its mean and standard deviation.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Estimation

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Question

Consider n independent random variables with gaussian probability densities but with
each having its own mean t; and standard deviation o;.
Find the probability density of the sum of those n variables as well as its mean and
standard deviation.

Expert Solution

Step 1

Given that;

$X_{i} ~ N o r m a l (μ_{i}, {σ_{i}}^{2}), i = 1, 2, . . ., n, i n d e p e n d e n t l y$

We know that the moment generating function of the normal distribution is given by;

$M_{X} (t) = e^{μ t + \frac{1}{2} {(t σ)}^{2}}$

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