‫ﺑﺴﻤﻪ ﺗﻌﺎﻟ‬
‫آﻣﺎر و اﺣﺘﻤﺎل ﻣﻬﻨﺪﺳ‬
٩۴-٩۵ ‫ﻧﯿﻢﺳﺎل اول‬
‫داﻧﺸ ﺪه ﻣﻬﻨﺪﺳ ﮐﺎﻣﭙﯿﻮﺗﺮ‬
‫ﺣﻤﯿﺪرﺿﺎ رﺑﯿﻌ ‐ ﺳﯿﺪﻋﺒﺎس ﺣﺴﯿﻨ‬
‫ آﺑﺎن‬١١ :‫زﻣﺎن ﺗﺤﻮﯾﻞ‬
‫ﺗﻤﺮﯾﻦ ﺳﻮم‬
1. Two last problems of 2nd homework (7&8).
2. Find density function of Y :
(a) X ∼ λ2 e−λ|x| , Y = e−α|X|
(b) x ∼ N (µ, σ 2 ), Y = eX
3.
(a) Prove P (X > a) ≤
inequality.
E[X]
a
for non-negative random variable X. This inequality is called Markov
(b) Using above inequality show that P (X > A) ≤ e−sA ϕ(s) for any s.
4. We toss a coin n times. Find the least n such that the probability that the number of tails is between 0.49n
and 0.51n is at least 0.9.
5. Bag 1 contains 8 white balls and 7 black ones. Bag 2 contains 5 white balls and 12 black balls. We take
3 balls from bag 1 and add them to bag 2. Then take 4 balls from bag 2. What is the mean of number of
white bags we have?
6. We have m blue pens and n red ones. We put them into a bag and take one of them randomly from it each
time until we see a red one. What is the mean of the number of the pens drawn until finding first red pen?
Give an explicit answer (and also not in infinite sum format).
7. There is a kind of game in a club that participant first draw a card from a box that gives cards with random
numbers on them. (On every card given by machine there is exactly one number.) Then he goes through
a very long hall. There are stations there that as he goes he should check his card number in each one.
At ith station he should check his card number with the number provided there, if his number is greater
he would get i dollars and continue passing the hall else he would lose the game. Specify a loss cost to
get form loser to have an equal game given that the number of the card drawn from the box has expected
value of 100 and variance 200.
8.
(a) Prove : V ar(X) = E[V ar(X|Y )] + V ar(E[X|Y ])
(b) Suppose a kangaroo that every second steps backward or forward with equal probability. Every day
is starts jumping until getting tired. It jumps around 1000 steps each day at mean. But it is not 1000
times each day. The number of jumps is a random variable with normal distribution with mean 1000
and variance 500. What is mean and variance of distance of the kangaroo after getting tired? (the
unit of distance is kangaroo steps)
١
٢