Problem 1A (expectation and p.d.f. of a function of a random variable, Y = g(X)). Consider a random variable X N Uniform[e, 82] and dene the new random variable Y = lnX. (a) Compute E(Y), using the theorem that states that E(g(X)) = If; g(a:)fX(:r) also. (b) Now we will calculate E(Y) by computing the probability density function of Y rst. To do so the initial step is to gure out what the c.d.f. of Y is, by noticing that: my) = my 5 y) = P(1nX : y) '2 M s all) = PM). (1) Note that in step (4:) we have applied the exponential function to both sides of the inequality, and used the fact that such function is monotone increasing (that is, a S b if and only if e\" S eb. Had we used a monotone decreasing function, we would have had to switch the direction of the inequality!). To nd the p.d.f. of Y apply the chain rule2 to (1). Now that you have fy, compute the expectation of Y via the usual formula E(Y) : If; yfy) dy. You should get the same result as in part (a)