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
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
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 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 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 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 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 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