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1. The RVs X1, X2, …, Xn, Y1, Y2, …, Yn are independent and U(0, a)-
distributed. Show that
Z ? n?log? max{X(n) ,Y(n)}?
has an Exp(1) distribution.
? min{X(n) ,Y(n)
?
2. Let X1, X2, … be independent random variables such that Xk?
Po(k), k = 1, 2, ?. Show that
1 ? n n
2 ?
Zn ? ??X k ? ?
n ?k?1 ?
converges to a N(½, ½) distribution an n ? .
3. Let Zn be defined as in Problem 2. Showthat
Zn ? 1 ( ?1)
Wn ? 2
converges in probability (and determine its limit).
4. Events occur according to a Poisson process with intensity ?. Each
time an event occurs, we must decide whether or not to stop, with our objective being to
stop at the last even to occur prior to some specified time T, where T > 1/?. That is, if an
event occurs at time t, 0 ? t ? T, and we decide to stop, then we win if there are no
additional events by time T, and we lose otherwise. If we do not stop when an event occurs
and no additional events occur by time T, then we lose. Also, if no events occur by time T,
then we lose. Consider the strategy that stops at the first event to occur after some fixed
time s, 0 ? s ? T.
a. Using this strategy, what is the probability ofwinning?
b. What value of s maximizes the probability of winning?
c. Using the value of s specified in part b., what is the probability ofwinning?
5. A system experiences a random number of flaws that occur according to a Poisson
process with intensity ?. Each of the flawswill, independently, cause the system to fail
at a random time having distribution function G. When a system failure occurs, the flaw
causing the failure is immediately located and fixed.
a. What is the distribution of the number of failures at time t?
b. What is the distribution of the number of flaws that remain in the systemat time t?
c. Are the random variables in parts a. and b. dependent orindependent?
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1. The RVs X , X , …, X , Y , Y , …, Y are independent and U(0, a)- 1 2 n 1 2 n distributed. Show that ? ? max{X ,Y } ? (n) (n) ? ?? Z ? n ? log n ? ? ? min{X ,Y } (n) (n) ? ???? has an Exp(1) distribution. ? 2. Let X , X , … be independent random variables such that X?? 1 2 k Po(k), k = 1, 2, ?. Show that 2 n 1 ? n?? Z? X ? ? ?? n ? k n 2 k ?1 ? ?? converges to a N(½, ½) distribution an n ? . 3. Let Z be defined as in Problem 2. Show that n 1 Z? ( ? 1) n n 2 W? n n converges in probability (and determine its limit). 4. Events occur according to a Poisson process with intensity ?. Each time an event occurs, we must decide whether or not to stop, with our objective being to stop at the last even to occur prior to some specified time T, where T > 1/?. That is, if an event occurs at time t, 0 ? t ? T, and we decide to stop, then we win if there are no additional events by time T, and we lose otherwise. If we do not stop when an event occurs and no additional events occur by time T, then we lose. Also, if no events occur by time T, then we lose. Consider the strategy that stops at the first event to occur after some fixed time s, 0 ? s ? T. a. Using this strategy, what is the probability of winning? b. What value of s maximizes the probability of winning? c. Using the value of s specified in part b., what is the probability of winning? 5. A system experiences a random number of flaws that occur according to a Poisson process with intensity ?. Each of the flaws will, independently, cause the system to fail at a random time having distribution function G. When a system failure occurs, the flaw causing the failure is immediately located and fixed. a. What is the distribution of the number of failures at time t? b. What is the distribution of the number of flaws that remain in the system at time t? c. Are the random variables in parts a. and b. dependent or independent?