SYS 6005 - Homework 9 solutions
1. Estimating the rate of a Poisson process.
Let Ti be the random variable giving the time of the i-th arrival. We observe the event
T1 = t1 , . . . , Tk = tk , Tk+1 > tmax , and we want to estimate the value of .
The inter
SYS 6005 - Homework 2 Solutions
1. Conditional expectation.
(a) As an example, suppose that X takes values in cfw_2, 1, 1, 2, each with probability 1 . Also, suppose that Y = X 2 . Clearly, knowing the value of Y conveys
4
some information about the value
Thus, the observation of a white cow makes the hypothesis all cows are white more
likely to be true.
Solution to Problem 1.27. Since Bob tosses one more coin that Alice, it is imu
possible that they toss both the same number of heads and the same number o
The event A ["1 B can be written as the union of two disjoint events as follows:
AHB=4AanCMAanU
so that
Pcfw_AHB=Pcfw_AHBHC]+P[AHBHCE. cfw_2]
Similarly,
P[AFIC=P(AHBHC]+Pcfw_AHHCHC]. (3]
Combining Eqs. (1)43), we obtain the desired result.
Solution to Pro
Let E (or E or E be the event that A cfw_or B or C, respectively) occurs and you first
select the envelope containing the larger amount . Let g (or E or Q) be the event
that A [or B or C, respectively) occurs and you rst select the envelope containing the
Solution to Problem 1.31. cfw_a Let A be the event that a [l is transmitted. Using
the total probability theorem, the desired probability is
Penn an + (1 Franc E11=a1 ea + (1 p1cfw_1 El)-
cfw_b By independence, the probability that the string lll is recei
The term ptupd cortesponds to the win~draw outcome, the term pw ppw corre-
sponds to the winulosenwin outcome, and the term cfw_1 pw p, corresponds to losenwinu
win outcome.
cfw_b If pm *5: 1,12, Boris has a greater probability of losing rather than winni
as likely if we know that B has occurred than if we know that C has occurred. Alices
reasoning corresponds to the special case where C = A U B.
Solution to Problemi 1.16. In this problem, there is a tendency to reason that since
the opposite face is eithe
and
AEHH=cfw_2, AEFIB:cfw_4,E, AHB=cfw_5.
Thus, the equality of part cfw_b is veried.
Solution to Problem 1.5. Let G and C be the events that the chosen student is
a genius and a chocolate lover, respectively. We have P09] : cfw_1.6, Pcfw_C] = DEF, and
P0
CHAPTER 1
Solution to Problem 1.1. We have
A:cfw_2,4,E, B=cfw_4,5,,
so A UB :cfw_2,1,5,, and
cfw_ALISE = cfw_1,3.
on the other hand,
A sec 2 cfw_1,3,3r1 cfw_1,2,3 2 cfw_1,3.
Similarly, we have A n B = cfw_4, 3, and
cfw_A r1 .3) = cfw_1, 2,3, 5.
on the oth
Figure 1.1: Sequential descriptions of the chess match histories under strategies
[1], (ii), and cfw_iii.
for e drew. In the case of e tied 1&1 score, we go to sudden death in the next game,
and Boris wins the match [prbbitf pm), or loses the match (proba
theorem. We have
1 1
Fri-mm g ' E 'Pn1.1cfw_3l=
n. 1 1 2 2
Fn1,1(3] pn,u[2) + 2- n - E -pn1.1[2 + E - H 'Pna):
1 1
Pa 0(21 g ' E 'PnI 1(1):
n 1 1
pn1,1(2] 1T ' E 'Pnmil]:
1 1
Ian2.21:2] L ' Finlull:
'11.
fin1.1]: 1
Combining these equations, we obtain
SYS 6005 - Homework 8 Solutions
Word recognition for your smartphone.
Using the Viterbi algorithm on the three inputs obtains:
this is a test
hello my hame is randu
this will work berttr with a dichionary
The Matlab code used to produce these outputs i
SYS 6005 - Homework 7 solutions
1. Expected total value of transitions.
Using iterated expectation, we have
E [ r(Xk , Xk+1 )] = E [ E[r(Xk , Xk+1 ) | Xk ] ]
We can dene the function g (i) = E[r(Xk , Xk+1 ) | Xk = i ], giving
E [ r(Xk , Xk+1 )] = E [ g (X
SYS 6005 - Homework 6 solutions
1. Modeling a waiting line.
(a) To simplify notation we will introduce the quantities
b1 = a (1 d )
b2 = d (1 a )
The transition matrix in terms of these quantities is
1 b1
b1
0
0
0
0
b2
1 b1 b2
b1
0
0
0
0
b2
1 b1 b2
b1
0
SYS 6005 - Homework 4 solutions
1. Estimating vehicle speed
(a) When conditioned on S = s, D is a normal random variable with conditional
mean
L + E
E[D | S = s] =
s
and conditional variance
2
2
L + E
var(D | S = s) =
s2
When substituting this mean and va
SYS 6005 - Homework 3 Solutions
1. Conditioning on a set of measure zero.
(a) Let S = cfw_x1 , . . . , xn . For some > 0, we will start by considering
P(X (xi , xi ] | X (x1 , x1 ] (xn , xn ]) =
P(X (xi , xi ])
P(X (x1 , x1 ] (xn , xn ])
For suciently sma
SYS 6005 - Homework 1 Solutions
1. Intersections of events.
(a) To provide an upper bound, suppose all 60% who were born in Virginia also like
tomatoes. In this case, 60% of the students both like tomatoes and were born in
Virginia.
To provide a lower bou