eet 𝑋 be a continuous random variable with PDF that is symmetric about zero, and let𝑌 = 𝑆𝑋 , where 𝑆 is a discrete random variable independent of 𝑋 and has PMF𝑃(𝑆 = 1) = 𝑃(𝑆 = −1) = 0.5. Compute Cov(𝑋, 𝑌). Are 𝑋 and 𝑌 independent? Justifyyour answer. (10 points

First, let's compute the covariance between X and Y. Covariance is a measure of how much two random variables vary together. It's similar to variance, but where variance tells you how a single variable varies, covariance tells you how two variables vary together.

The Covariance is given by the formula:

Cov(X,Y) = E[XY] - E[X]E[Y]

Given that Y = SX, we can substitute Y in the formula:

Cov(X,Y) = E[X(SX)] - E[X]E[SX]

Since S and X are independent, we can simplify E[X(SX)] to E[S]E[X^2]. And because the PDF of X is symmetric about zero, E[X] = 0.

So, Cov(X,Y) = E[S]E[X^2] - 0E[SX] = 0, because E[S] = 0.51 + 0.5*(-1) = 0.

Therefore, the covariance of X and Y is 0.

However, zero covariance does not imply independence. Covariance measures linear dependence between variables. Two variables could be dependent in other ways and yet have zero covariance.

In this case, X and Y are not independent. This is because the value of Y is determined by X (since Y = SX), so knowing the value of X gives information about Y, and vice versa. Therefore, X and Y are dependent.