Worksheet 5 Solutions
Please tell me if you find any errors in these solutions!
1. Consider the set of rectangles with perimeter P . Show that the rectangle in this set with
the biggest area is a square.
Solution: Call the side lengths a and b. Then 2a +
Worksheet 3 Solutions
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d
1. Compute the derivatives dr
of the following functions of r. Here a is a constant, and f is
an unknown function with f 0 (u) = g(u).
(a) (r2 + r)er
(b) (r + ar3/4 )er
(c)
Worksheet 4 Solutions
Please tell me if you find any errors in these solutions!
1. Where on the curve cos(xy) = ex is the tangent line vertical?
Solution: We want to know when
dx
dy
= 0. Applying
d
dy
to both sides of the equation,
dx
dx
y = ex
sin(xy) x
Worksheet 1 Solutions
Please tell me if you find any errors in these solutions!
1. Evaluate the limit lim
x
Solution:
p
x2 x + x or show that it does not exist.
p
x2 x x
p
2
2
lim
x x + x = lim
x x+x
x
x
x2 x x
x2 x x2
= lim
x x2 x x
x
= lim p
x x2 1 1
Worksheet 1 Solutions
Please tell me if you find any errors in these solutions!
1. Solve for x:
32x + 3x 1/2 = 0
Solution: Secretly this is a quadratic equation in 3x :
(3x )2 + 3x 1/2 = 0
Applying the quadratic formula,
p
1 (4)(1)(1/2)
3 =
2
1 3
x
3 =
2
Solutions
180B FINAL
Please simplify your answers to the extent reasonable without a calculator, show your
work, and explain your answers, concisely. If you set up an integral or a sum that you
cannot evaluate, leave it as it is; and if the result is need
180B Extra Credit 1 (due Monday, February 13, 2017)
Please turn in separately from your regular homework, i.e., on a separate piece of paper.
EC1. Problem 1 on the midterm involved a Markov chain with two weather states, Wt
cfw_S, R, where the t N index
Solutions
180B PRACTICE MIDTERM 2
Please simplify your answers to the extent reasonable without a calculator. Show your
work. Explain your answers, concisely.
1. Two bugs start at opposite corners of a square. During each minute they each walk
along an ed
180B PRACTICE MIDTERM 2
Please simplify your answers to the extent reasonable without a calculator. Show your
work. Explain your answers, concisely.
1. Two bugs start at opposite corners of a square. During each minute they each walk
along an edge, chosen
Math 180B Problem Set 3
Problem 1. (Exercise 3.1.2)
Solution. By the definition of conditional probabilities we have
Prcfw_X2 = 1, X3 = 1 | X1 = 0 = Prcfw_X3 = 1 | X2 = 1, X1 = 0 Prcfw_X2 = 1 | X1 = 0
= P1,1 P0,1 = .6 .2 = .12.
Note the second equality us
Solutions
180B MIDTERM 1
Please simplify your answers to the extent reasonable without a calculator. Show your
work. Explain your answers, concisely.
1. Especially in San Diego, a good guess about tomorrows weather is that it will be like
todays. Lets for
Solutions
180B PRACTICE MIDTERM 1
Please simplify your answers to the extent reasonable without a calculator. Show your
work. Explain your answers, concisely.
1. Consider the following game: A player rolls a die with the numbers 1, . . . , 6 on its faces.
180B PRACTICE MIDTERM 1
Please simplify your answers to the extent reasonable without a calculator. Show your
work. Explain your answers, concisely.
1. Consider the following game: A player rolls a die with the numbers 1, . . . , 6 on its faces.
If it sho
180B HWK 3 (due Monday, January 30, 2017)
Ex. 3.1.2, 3.1.5; Pr. 3.1.2
Ex. 3.2.2, 3.2.6; Pr. 3.2.2, 3.2.5
Ex. 3.3.1; Pr. 3.3.2, 3.3.7
1. Suppose X, Y have a bivariate normal distribution with X N (0, 1) and Y |X = x
N (ax + b, 2 ).
a. What is the marginal
Math 180B
(P. Fitzsimmons)
Second Midterm Exam
Solutions
1. A taxicab driver moves back and forth between the airport A and two hotels B and
C according to the following rules. If he is at the airport, he will go to one of the two
hotels next, with equal
Math 180B, Winter 2015
Notes on covariance and the bivariate normal distribution
1. Covariance.
If X and Y are random variables with nite variances, then their
covariance is the quantity
(1.1)
Cov(X, Y ) := E[(X X )(Y Y )],
where X = E[X] and Y = E[Y ]. T
Math 180B
(P. Fitzsimmons)
First Midterm Exam
Solutions
1. Let X and Y be random variables with Var(X) = 1, Var(Y ) = 4, and Var(X + Y ) = 4.
Find Cov(X, Y ) and Corr(X, Y ).
Solution. Lets compute:
4 = Var(X + Y ) = Var(X) + Var(Y ) + 2 Cov(X, Y ) = 1 +
Math 180B
Introduction to Stochastic Processes, I
Winter 2015
This course is an introduction to some basic topics in the theory of Stochastic Processes. After
nishing the discussion of multivariate distributions and conditional probabilities initiated in
Math 180B
(P. Fitzsimmons)
Second Midterm Exam
Solutions
1. A bug moves at random on the state space cfw_0, 1, 2 as follows: From state 0 the bug moves to either 1
or 2, each with probability 1/2; from state 1 the bug moves to 2 with probability 1/3 and r
Math 180B, Winter 2015
Homework 2 Solutions
1. Use the formula P(A) = P(A|B)P(B) + P(A|B c )P(B c ) to prove that if P(A|B) = P(A|B c )
then A and B are independent. Then prove the converse (that if A and B are independent then
P(A|B) = P(A|B c ).
Solutio
Math 180B, Winter 2015
Homework 1 Solutions
1. A fair coin is tossed 20 times. Let X be the number of heads thrown in the rst 10
tosses, and let Y be the number of heads tossed in the last 10 tosses. Find the conditional
probability that X = 5, given that
Math 180B, Winter 2015
Homework 2
1.
Use the formula P(A) = P(A|B)P(B) + P(A|B c )P(B c ) to prove that if P(A|B) =
P(A|B c ) then A and B are independent. Then prove the converse (that if A and B are
independent then P(A|B) = P(A|B c ).
2. Let X1 and X2
Math 180B, Winter 2015
Homework 1, due January 15
1. A fair coin is tossed 20 times. Let X be the number of heads thrown in the rst 10
tosses, and let Y be the number of heads tossed in the last 10 tosses. Find the conditional
probability that X = 5, give
Math 180B, Winter 2015
Homework 7 Solutions
Ex. 4.2.2. In this example, the repair facility is idle precisely when the system is in state
q2
(2, 0). The long run frequency of this state (see page 180) is 0 = 1+p+p2 . When a second
repair facility is added
Math 180B, Winter 2015
Homework 5 Solutions
Ex. 3.5.2. (a) If p = .49292929 then q/p = 1.028688537, and by formula (5.13) on page 144, the
probability that A is ruined before B is
(1.028688537)50 (1.028688537)100
= 0.804433.
1 (1.028688537)100
If each pla
Math 180B, Winter 2015
Homework 6 Solutions
Ex. 4.1.1. This transition matrix is clearly regular, so to nd the limit distribution we
have only to nd the stationary distribution (see Theorem 4.1, page 168). That is, we
need to solve
[ 0
1
.7
0
2 ]
.5
.1
.
Math 180B, Winter 2015
Homework 8 Solutions
Ex. 4.4.1. The stationary distribution equation P = yields
k = pk1 ,
k = 1, 2, 3, 4.
k = p k 0 ,
k = 1, 2, 3, 4.
Therefore,
But
4
4
k = 0
1=
k=0
1 p5
p = 0
,
1p
k
k=0
so 0 = q/(1 p5 ), and
k = qpk /(1 p5 ),
k =
Math 180B, Winter 2015
Homework 4 Solutions
Ex. 3.3.2.
For k = 0, 1, 2, . . . , N 1, a k k + 1 transition occurs if the chosen ball
comes from urn B and then urn A is chosen to receive it; this happens with probability
(1 k/N )p. Likewise, for k = 1, 2, .
Math 180B, Winter 2015
Homework 3 Solutions
Ex. 3.1.4.
P[X1 = 1, X2 = 1|X0 = 0] = P0,1 P1,1 = .1 .2 = .02,
and P[X2 = 1, X3 = 1|X1 = 0] has the same value because the transition probabilities do not depend on the
absolute time:
P[X1 = 0, X2 = 1, X3 = 1]
P