Question

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.

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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(a) Show that Y ∼ expon(2λ)
(b) Find E(Y ) and E(Z).
(c) Find the conditional density fZ|Y (z|y).
(d) FindP(Z>2Y).

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X1, X2, X3, and X4 and
Y1, Y2, Y3, Y4, and
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(A) N2
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also suppose the X2 is normally distributed with mean 0 also
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find P(Y>=1) ?

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(c) cov(X1 , Y1 )
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b) Compute P(Y > 1).

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a) Use distribution function method to find the probability
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b) Compute P(Y > 1).
c) Compute E(Y )

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