Download Introduction to Probability MSIS 575 Homework 4

Introduction to Probability
MSIS 575
Homework 4
Instructor: Farid Alizadeh
Due Date: Saturday December 16, 2000 in class
last updated on December 14, 2000
1. Consider the following chain (If sum of probabilities out of a state does
not add up to 1 assume a loop over that state with probability equal to
the slack):
3
6
A
AK
A
A1
1
2/3
A1
A
1/3
A
A
A
AA
U
?
/
2
1
4
- 5
@
6
I 3/5
1
1
1/5
@
@ @
0
(a) Which classes are transient and which are persistent?
(b) Find the period of each state.
(c) What are the equivalence classes under the “communicate” relationship? Which group of classes are closed?
(d) Starting from 0, what is the probability of hitting 6?
(e) Starting from 1, what is the probability of hitting 3? What is the
average number of steps to to hit 3?
(f) Starting from 1, what is the long-term proportion of time spent in
each of states 2, 3 and 1?
2. A fair die with six sides, marked by numbers 1 through 6, is thrown
repeatedly. At the kth toss we are in state j if j is the largest number that
has occurred so far.
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MSIS 565, Fall 2000
Homework 4
Due date: 12/16/00
(a) What is the transition probability matrix of this chain?
(b) Which states are persistent and which are transient?
(c) What is the stationary distribution of this matrix?
3. During period n Zn+1 people enter a line for service in a bank (the period
is a fixed amount of time, say 30 seconds.) And on period n + 1 exactly
one person is served. Therefore denoting by Xn the number of people
waiting in line at period n we have,
Xn+1 = (Xn + 1)+ + Zn+1
where a+ = max(0, a). The random variables Zn are i.i.d with common
distribution pk = Pr[Z1 = k].
(a) Show that {Xn }n≥0 is a Markov chain with transition probability


p0 p1 p2 p3 · · ·
p0 p1 p2 p3 · · · 


 0 p0 p1 p2 · · · 


0
p0 p1 · · · 


..
..
..
..
.
.
.
.
(b) Optional bonus question: Let q = (q0 , q1 , q2 , . . . ) be the stationary distribution of the system. Remember that qT P = qT . If
the Generating function P(z) = p0 + p1 z + · · · is known, find the
generating function Q(z) = q0 + q1 z + · · · .
(c) Optional bonus question: If Zn each have Poisson distribution of
rate λ (that is the number of people entering the line at each period
follows the Poisson distribution), find the stationary distribution of
the chain.
4. If X is N(µ, σ), show that Pr[|X − µ| < 0.675σ] = 0.5.
5. If U is uniform on [0,1] what is the density √function of
distribution function, mean, and variance of U.
√
U? Also find
6. If U is uniform in [-1,1] what is density function of U2 ? Also find distribution function, mean, and variance of U2 .
7. Find the density function of eZ if Z is N(µ, σ). Find Distribution, mean
and variance o eZ . (This distribution is called the lognormal distribution
since its logarithm is normal.)
8. Note: You may wish to start working on this question after
Mondays’s lecture. The joint density function of Random variables X
and Y is given by
2 −λy
λ e
, 0 ≤ x ≤ y, λ > 0
f(x, y) =
0
elsewhere
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MSIS 565, Fall 2000
Homework 4
Due date: 12/16/00
(a) Find the marginal density fX (x).
(b) Find the marginal density fY (y).
(c) Find the conditional density fX|Y (x|y). Also find E[X|Y = 2].
(d) Find the conditional density fY|X (y|x). Also find E[Y|X = 3].
9. canceled Show that if X is Fn,m then X−1 is Fm,n .
10. canceled Show that if T is tn then T 2 is F1,n .
11. canceled If X and Y are independent exponential random variables with
λ = 1, then X/Y has F distribution. Also find degrees of freedom m and
n.
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