Question

1.8. Expectation of a Random Variable 67 1.8.8. A bowl contains 10 chips, of which 8 are marked $2 each and 2 are $5 each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation. Let f(z) = 2z, o < 1, zero elsewhere, be the pdf of X. (a) Compute E(1/X). (b) Find the odf and the pdf of Y - 1/x. (c) Compute E(Y) and compare this result with the answer obtained in part (a) 1.8.10. Two distinct integers are chosen at random and without replacement from the first six positive integers. Compute the expected value of the absolute value of the difference of these two numbers. Let X have a Cauchy distribution which has the pdf 1.8.8) Then X is symmetrically distributed about 0 (why?). Why isn't E(X) ? 15.12Let X have the pf e)0ero elsewhere (a) Compute E(X3) (b) Show that Y X3 has a uniform(0, 1) distribution. (c) Compute E(Y) and compare this result with the answer obtained in part (a). 1.8.13. Using the probabilities discussed in Example 1.8.9 and independence, de- termine the distribution of the random variable G, the gain to a player of the game when he pays po dollars to play. Show that E(G)-$3.54 if the player pays $5 to play 1.8.14. A bowl contains five chips, which cannot be distinguished by a sense of touch alone. Three of the chips are marked $1 each and the remaining two are marked S4 each. A player is blindfolded and draws, at random and without replace- ment, two chips from the bowl. The player is paid an amount equal to the sum of chins that he draws and the game is over. Suppose it costs in to a player of

Answer 1

$=\int_{0}^{y^{1/3}}3x^2dx=y,~0<y<1\\ pdf~of~Y~is:\\ f(y)=\frac{\mathrm{d}F_Y(y) }{\mathrm{d} y}=1~for~0<y<1\\ =0~otherwise\\ c.~E(Y)=\int_{0}^{1}ydy=\frac{1}{2}$