Question: Let X be a uniform random variable on [0,1] and conditional on X=x. Let Y be an exponential random variable with mean x. a) find

Let X be a uniform random variable on [0,1] and conditional on X=x. Let Y be an exponential random variable with mean x.

a) find the joint density of X and Y

b) find E[Y]. (for this one the answer is 1/2, but I want know how to get it, please be detail )

c) find Var[Y].(answer is 5/12)

d)find cov(X,Y).(answer is 1/12)

Please show every step

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