Hi. Solve the following. Strictly answer every question

5-57. Suppose that X has a standard normal distribution. Let the conditional distribution of Y given X = x be normally dis- tributed with mean E(Y | x) = 2x and variance V(Y | x ) = 2x. Determine the following. (a) Are X and Y independent? (b) P(Y 2. (Recall (z) = z if z 2 0 and |z| = -z if z R be real functions. Consider the function / : R R defined by where @ is the set of rational numbers. Show that if / and g are continuous functions at a and (to) - gian), then h is continu1. (a) Use the Intermediate Value Theorem to show that the equation - sin (x) + sin(x) =-1 has at least two distinct real solutions on (0, ce). Justify your answer. (b) Let ( fa)=2 be the sequence of functions defined by: 1 -nex, fu( x) = 0. 2 it xe [ 3. 1 ] (1) Find the pointwise limit of the sequence of functions (.),2 Does the sequence ( ,), , converge uniformly on [0. 1| to its pointwise limit? Justify your answers Let a. B E N. and f : 8 - R be the function defined by: B (sin (;))" cos (!). f( x) = For cr EN and B > 2, show that function / is differentiable on R. and give the value of f(x) for allre R. (1) For de N and B - 1, show that / is not differentiable at 0