# Question: A pair of random variables X Y is equally likely

A pair of random variables, (X, Y), is equally likely to fall anywhere within the region defined by |X| + |Y| ≤ 1.

(a) Write the form of the joint PDF, fX,Y (x,y).

(b) Find the marginal PDFs, fX (x) and fY (y).

(c) Find and Pr (X > 1 / 2) and Pr (Y < 1 / 2)

(d) Find Pr (Y < 1 / 2|X > 1/ 2). Are the events {X > 1 / 2} and {Y < 1 / 2} independent?

(a) Write the form of the joint PDF, fX,Y (x,y).

(b) Find the marginal PDFs, fX (x) and fY (y).

(c) Find and Pr (X > 1 / 2) and Pr (Y < 1 / 2)

(d) Find Pr (Y < 1 / 2|X > 1/ 2). Are the events {X > 1 / 2} and {Y < 1 / 2} independent?

## Answer to relevant Questions

For some integer and constant , two discrete random variables have a joint PMF given by (a) Find the value of the constant in terms of L. (b) Find the marginal PMFs, P M (m) and PN (n). (c) Find Pr (M + N < L / 2). Consider again the random variables of exercise 5.12 that are uniformly distributed over an ellipse. (a) Find the conditional PDFs, fX|Y (x| y) and fY|X (y|x). (b) Find f X|Y > 1(x). (c) Find fY |{|X| < 1}. Prove the triangle inequality which states that Find an example (other than the one given in Example 5.15) of two random variables that are uncorrelated but not independent. Suppose a random variable X has a CDF given by Fx (x) and similarly, a random variable Y has a CDF, Fy (y) . Prove that the function F(x,y) = Fx (x) Fy (y) satisfies all the properties required of joint CDFs and hence will ...Post your question