196 Chapter 5: Special Random Variables
turning over the fourteenth card, and so on.) You lose if you ever turn over a card
that matches what you have just said. Use the Poisson paradigm to approximate
the probability of winning. (The actual probability is .01623.)
16.Theprobabilityoferrorinthetransmissionofabinarydigitoveracommunication
channel is 1/10^3. Write an expression for the exact probability of more than 3
errors when transmitting a block of 10^3 bits. What is its approximate value?
Assume independence.
17.IfXis a Poisson random variable with meanλ, show thatP{X=i}first increases
and then decreases asiincreases, reaching its maximum value wheniis the largest
integer less than or equal toλ.
18.A contractor purchases a shipment of 100 transistors. It is his policy to test 10
of these transistors and to keep the shipment only if at least 9 of the 10 are in
working condition. If the shipment contains 20 defective transistors, what is the
probability it will be kept?
19.LetXdenote a hypergeometric random variable with parametersn, m, andk.
That is,P{X=i}=(n
i)( m
k−i)(
n+m
k) , i=0, 1,..., min(k,n)(a)Derive a formula forP{X=i}in terms ofP{X=i− 1 }.
(b) Use part (a) to computeP{X=i}fori=0, 1, 2, 3, 4, 5 whenn=m=10,
k=5, by starting withP{X= 0 }.
(c) Based on the recursion in part (a), write a program to compute the
hypergeometric distribution function.
(d) Use your program from part (c) to computeP{X≤ 10 }whenn=m=30,
k=15.
20.Independent trials, each of which is a success with probabilityp, are successively
performed. LetXdenote the first trial resulting in a success. That is,Xwill equal
kif the firstk−1 trials are all failures and thekth a success.Xis called ageometric
random variable. Compute
(a)P{X=k},k=1, 2,...;
(b) E[X].
LetY denote the number of trials needed to obtainrsuccesses.Y is called a
negative binomial random variable. Compute
(c) P{Y=k},k=r,r+1,....