1.8. Expectation of a Random Variable 671.8.8.A bowl contains 10 chips, of which 8 are marked $2 each and 2 are marked
$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.1.8.9.Letf(x)=2x, 0 <x<1, zero elsewhere, be the pdf ofX.(a)ComputeE(1/X).(b)Find the cdf and the pdf ofY=1/X.(c)ComputeE(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.
1.8.11.LetXhave a Cauchy distribution which has the pdf
f(x)=1
π1
x^2 +1
, −∞<x<∞. (1.8.8)ThenXis symmetrically distributed about 0 (why?). Why isn’tE(X)=0?1.8.12.LetXhave the pdff(x)=3x^2 ,0<x<1, zero elsewhere.(a)ComputeE(X^3 ).(b)Show thatY=X^3 has a uniform(0,1) distribution.(c)ComputeE(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 variableG, the gain to a player of the game
when he paysp 0 dollars to play. Show thatE(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 $4 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
the values of the two chips that he draws and the game is over. Suppose it costs
p 0 dollars to play the game. Let the random variableGbe the gain to a player of
the game. Determine the distribution ofGand theE(G). Determinep 0 so that
the game is fair. The R codesample(c(1,1,1,4,4),2)computes a sample for this
game. Expand this into an R function that simulates the game.