1) Consider a pair of random variables X and Y. a) Show that Mm) = 9:2 and 9(m) = am for some a 6 R are measurable functions. [2 points] b) Use part (a) and the fact that sums of random variables are random variables to show that X Y is a random variable. [3 points] [I-Iint: Note that ab = W] ii coursehero.com E u Find study resources O\\ a) Consider a pair of random variables X and Y. We will show that: h(x)=x2 and g(x)=ax for some a R are measurable functions. We have the following relations: h(X)=X2;g(X)=AX. [Y is not needed to derive this]Therefore, f (g (X))=AX*. Now, if h()()=X2 and g(X)=AX, then f (g (X)) =2AX*. Explanation: Consider the equation: f(x) = ax. This implies that f(x) is a commutative function (a constant). So, this means X and Y are independent random variables because 900 is a function of X. Therefore, for each value of X, we have a different value for Y. We know that: f (g (X)) = f (AX) = aX2. Therefore, f(g(X)) = a* X2 and so g(x) is measurable. If f(g(X))=2AX*, then, by definition of *, f((g(x)a) equals g((x*a), and therefore 9 (x*) is measurable. Therefore, 9 (x*) exists for all x in the domain of h. b) Use part (a) and the fact that sums of I 'II I 'II I b) Use part (a) and the fact that sums of random variables are random variables to show that XY is a random variable. The Y random variable is defined by the formula Y = h(X+g(x)) where X and g are as defined in part (a). Since Y = f(X+g) and f is continuous, then Y has probability mass at every point in its domain. Since the domain of Y is open, we can conclude that Y is an open set. This justifies the statement that XY is a random variable