Question

Let (X1, X2) have joint pdf f(x1, x2) = (2/9)x1x22, 0 <= x1 <= 1, 0...

Let (X1, X2) have joint pdf

f(x1, x2) = (2/9)x1x22, 0 <= x1 <= 1, 0 <= x2 <= 3

(i) What is the distribution of Y = X1 + X2?

(ii) What is the distribution of Y = X1 * X2?

(iii) Find the expectation E(X1 + X2)

(iv) Find the expectation E(X1X2)

Math   Statistics-And-Probability   
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Answer #1

The joint probability distribution is:

We can easily see that x1 and x2 are separable and therefore independent variables. The probability distributions for X1 and X2 are

We can verify that the area under both the random variables is 1.

(i) As they are independent, the distribution of Y = X1 + X2 can be obtained by convolution.

(iI) The distribution of product of the independent random variables is given by:

To find the expected values, we have to find expected values of each random variable first.

So, E(Y) = E(X1) + E(X2)

(iv) E(Y) = E(X1)*E(X2)

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