Question

Suppose that X1 and X2 are independent standard normal random variables. Show that Z = X1...

Suppose that X1 and X2 are independent standard normal random variables. Show that Z = X1 + X2 is a normal random variable with mean 0 and variance 2.

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Answer #1

Given:

X1 is Standard Normal Variate. So, Probability Density Function of X1 is given by:

X2 is Standard Normal Variate. So, Probability Density Function of X2 is given by:

The Probability Density Function of

Z = X1 + X2 is given by Convolution Theorem as follows:

The expression in the brackets = 1, since it is the integral of the Normal Density Function with = 0 and = .

So,

we have:

This proves the required result that Z = X1 + X2 is a normal random variable with mean 0 and variance 2.

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