5. Do example 1.17 the long way: first find the pdf of V (note that the function H(x) = x3 is monotonically increasing) and then use the definition of expected value to find E[V].Example 1.17: Draw a line segment of lengthX, where X is a random variable with probability density function f (x) = 2x, 0 S x S 1. Now construct a cube whose side is X. What is the expected volume of the cube? l The volume is V = X3, so we seek E[V]. By LOTUS, [SW] = 1x3 . 2xdx = g. 0 It is important to realize that E[H(X)] at H(E[X]). (One of the reasons the theorem is called the Law of the Unconscious Statistician is that, allegedly, some feeble-brained statisticians believe this is true.) So, for instance, EI._JEI1X 42 ]and E[J_] #J EIX]. Thus, we cannot nd the expected volume In Example 1.17 by X]E[V] = (E[X])3 = (2)13 3 27