Let the random variable X and Y have joint distributionP (X = a, Y = 0) = P (X = 0, Y = a) = P (X = −a, Y = 0) = P (X = 0, Y = −a) = 1/4.Show that X − Y and X + Y are independent.

To show that X - Y and X + Y are independent, we need to show that the joint probability distribution of X - Y and X + Y is equal to the product of their marginal distributions.

Let's denote Z = X - Y and W = X + Y. We need to find P(Z, W) and show that it equals P(Z)P(W).

First, let's find the possible values of Z and W:

Z = X - Y can be a, -a, -a, a (corresponding to the pairs (a, 0), (0, a), (-a, 0), (0, -a)) W = X + Y can be a, a, -a, -a (corresponding to the same pairs)

Now, let's find P(Z, W):

P(Z = a, W = a) = P(X = a, Y = 0) = 1/4 P(Z = -a, W = a) = P(X = 0, Y = a) = 1/4 P(Z = -a, W = -a) = P(X = -a, Y = 0) = 1/4 P(Z = a, W = -a) = P(X = 0, Y = -a) = 1/4

Now, let's find P(Z) and P(W):

P(Z = a) = P(X = a, Y = 0) + P(X = 0, Y = -a) = 1/4 + 1/4 = 1/2 P(Z = -a) = P(X = 0, Y = a) + P(X = -a, Y = 0) = 1/4 + 1/4 = 1/2 P(W = a) = P(X = a, Y = 0) + P(X = 0, Y = a) = 1/4 + 1/4 = 1/2 P(W = -a) = P(X = 0, Y = -a) + P(X = -a, Y = 0) = 1/4 + 1/4 = 1/2

Finally, let's check if P(Z, W) = P(Z)P(W):

P(Z = a, W = a) = 1/4 = P(Z = a)P(W = a) = 1/2 * 1/2 = 1/4 P(Z = -a, W = a) = 1/4 = P(Z = -a)P(W = a) = 1/2 * 1/2 = 1/4 P(Z = -a, W = -a) = 1/4 = P(Z = -a)P(W = -a) = 1/2 * 1/2 = 1/4 P(Z = a, W = -a) = 1/4 = P(Z = a)P(W = -a) = 1/2 * 1/2 = 1/4

Since P(Z, W) = P(Z)P(W) for all possible values of Z and W, we can conclude that X - Y and X + Y are independent.