Two individuals, A and B, both require kidney transplants. If she does not recei

Question

Two individuals, A and B, both require kidney transplants. If she does not receive
a new kidney, then A will die after an exponential time with rate A, and B after an
exponential time with rate B. New kidneys arrive in accordance with a Poisson
process having rate . It has been decided that the first kidney will go to A (or to
B if B is alive and A is not at that time) and the next one to B (if still living).

(a) What is the probability that A obtains a new kidney?
(b) What is the probability that B obtains a new kidney?
(c) What is the probability that neither A nor B obtains a new kidney?
(d) What is the probability that both A and B obtain new kidneys?

Explanation / Answer

a)The event that A obtains a new kidney happens when the first kidney arrives before A dies; i.e.,with Tk denoting the arrival time of the k th kidney,

if T1 < TA, so P(A obtains a new kidney) = P(T1 < TA) = / + µA

b)To obtain the probability of B obtains a new kidney, we will condition on the first event (and then uncondition), i.e., which happens first: a kidney arrives, A dies or B dies.

Thus P(B obtains a new kidney) = P(B obtains a new kidney|T1 = min {T1, TA, TB})

P(T1 = min {T1, TA, TB}) +P(B obtains a new kidney|TA = min {T1, TA, TB})

P(TA = min {T1, TA, TB}) +P(B obtains a new kidney|TB = min {T1, TA, TB})

P(TB = min {T1, TA, TB}) = P(T2 < TB)P(T1 = min {T1, TA, TB}) + P(T1 < TB)P(TA = min {T1, TA, TB}) + 0

= ( / + µB )( / + µA + µB )´ + ( / + µB )(µA / + µA + µB )

= ( / + µB )( + µA / + µA + µB ).

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.