# Discussing the Distribution Theory

1. Let the random variable X have the pdf

f(x) = 2/sqrt(2(pi)) e^(-x^2/(2) ), 0<x<infinity, zero elsewhere.

Find the mean and the variance of X.

Hint: Compute E(X) directly and E(X^2) by comparing the integral with the integral representing the variance of a random variable that is N(0,1).

2. Compute P(X_1 + 2X_2 - 2X_3 > 7), if X_1, X_2, X_3 are iid with common distribution N(1,4).

3. Let X and Y have a bivariate normal distribution with parameters u1 = 20, u2 = 40, sigma1^2 = 9, sigma2^2 = 4, and p = 0.6. Find the shortest interval for which 0.90 is the conditional probability that Y is in the interval, given that X = 22.

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#### Solution Preview

Solution 1

It is given that,

We have,

Put , then and as x varies from 0 to , u also varies from 0 to .

Thus,

Also,

.

We know that, for a Standard normal random variable U,

That is, (1)

Since the integrand is an even function , and hence (1) becomes ...

#### Solution Summary

This solution illustrates how to solve a number of distribution theory problems and the response is enclosed within an attached Word document.