Question

Please Only Do Question 2

[1] The joint probability density function of two continuous random variables X and Y is fxxx(x,y) = {S. sc, 0 <y s 2.y = x < 4-y otherwise Find the value of c and the correlation of X and Y. [2] Consider the same two random variables X and Y in problem [1] with the same joint probability density function. Find the mean value of Y when X<1.

Answer 1

We are to perform the second question here using the data from first question

Q2) The mean of Y given X < 1 is computed here using Bayes theorem as:

Note that as 0 <= y < = 2, therefore 4 - y is always greater than 1.

Therefore, we can get the required mean value here as:

$E(Y | X < 1) = \frac{E(Y , X < 1)}{P(X < 1)}$

$E(Y | X < 1) = \frac{\int_{0}^{1}\int_{y}^{1} cy \ dx \ dy}{\int_{0}^{1}\int_{y}^{1} c \ dx \ dy }$

$E(Y | X < 1) = \frac{\int_{0}^{1} y(1-y) \ dy}{\int_{0}^{1}(1-y) \ dy }$

$E(Y | X < 1) = \frac{\frac{1}{2} - \frac{1}{3}}{1 - \frac{1}{2}} = \frac{1}{3}$

Therefore 1/3 is the conditional expected value of Y given X < 1 here.