Math 180C, Spring 2006
Homework 1 Solutions
Ex. 6.1.1. P0 (t) = et .
t
t
e3s et ds
e3(ts) P0 (s) ds = e3t
P1 (t) =
0
0
t
e2s ds = e3t (e2t 1)/2
= e3t
0
1
1
= et e3t .
2
2
t
e2(ts) P1 (s) ds =
P2 (t) = 3
0
3
2
t
e2(ts) (es e3s ) ds
0
3
3 2t t s
e
(e es ) d
Math 180C
(P. Fitzsimmons)
Midterm Exam
Solutions
1. Consider a birth-and-death process X(t) with state space cfw_0, 1, 2, 3, birth rates k = 2 for
k = 0, 1, 2, and death rates k = k for k = 1, 2, 3 (0 = 3 = 0). Let T0 = mincfw_t > 0 : X(t) = 0.
(a) Calcu
Math 180C, Spring 2013
Homework 6 Solutions
Ex. 7.5.1. One cycle for the server consists of two pieces: the idle time spent waiting for the
next customer (an exponential random variable with rate and expectation 1/) and the service
time of that customer (
Math 180C, Spring 2013
Homework 9 Solutions
Pr. 8.3.4. For 0 < s t < 1, we also have s/(1 s) t/(1 t), and so
E[W o (s)W o (t)] = (1 s)(1 t)E[B(s/(1 s)B(t/(1 t)]
= (1 s)(1 t)(s/(1 s)
= s(1 t),
which coincides with the covariance for the Brownian bridge.
Pr
Math 180C, Spring 2013
Homework 8 Solutions
Ex. 8.2.6. Evidently, cfw_1 < t if and only if the Brownian motion has a zero in the time
interval (b, t). According to formula (8.25) (page 408), the probability of this event is (b, t) =
2
b/t.
arccos
Pr. 8.2
`
Math 180C, Spring 2013
Homework 7 Solutions
Ex. 8.1.6. (a) Because B(u) and B(u + v) have a bivariate normal distribution, the random
variable B(u) + B(u + v) is normally distributed with mean
E[B(u) + B(u + v)] = E[B(u)] + E[B(u + v)] = 0 + 0 = 0
and v
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION
The length of a certain type of battery is normally distributed
with mean 5.0cm and standard deviation .05cm. Find the
probability that such a battery has a length between 4.92 and
5.08 c
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION
A man travels to work by train and bus. His train is due to
arrive at 08.45 and the bus he hopes to catch is due at 08.48.
The time of arrival of the train has a normal distribution with
QUESTION The time to failure,t in hours, of components follows the
density function f (t) = 2 tet, t > 0.
(a) What is the probability that a component which has survived for two
hours will fail in the next hour?
(b) The cost of producing a component is 22
QUESTION A continuous random variable X is uniformally distributed
in the interval 1 x 1. Find E(X) and Var(X). The random variable
Y
is defined by Y = X 2 , Use the cdf of X to show that P (Y y) = Y , o
y 1 and obtain the pdf of Y. Hence or otherwise ev
QUESTION The variable X has pdf f (x) = 18 (6 x) 2 x 6. A
sample of two values of X is taken. Denoting the lesser of the two values by
Y, use the cdf of X to write down the cdf of Y. Obtain the pdf of Y and the
mean of Y. Show that its median is approxima
QUESTION A man travels to work by train and bus. His train is due to
arrive at 08.45 and the bus he hopes to catch is due at 08.48. The time of
arrival of the train has a normal distribution with mean 08.44 and standard
deviation 2 mins; the departure tim
QUESTION A random variable X has age specific failure rate function
(x) = x. Find f(x) and F(x) and derive the mode and median of the
distribution of x.
R
R
1 2
ANSWER G(x) = e (x) dx = e x dx = e 2 x
1 2
f (x) = (x)G(x) = xe 2 x
1 2
F (x) = 1 G(x) = 1 e
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION The length of time a customer is in a queue waiting to be
served
at a certain cash point has cdf $F(x)=1-pe^cfw_-\lambda x, x \geq
0, \lambda >0,0<p<1$. Find P(x=0) and the pdf for x$>$0.
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION
The variable X has pdf $f(x)=\fraccfw_1cfw_8(6-x)\ \ 2 \leq x \leq
6$. A sample of two values of X is taken. Denoting the lesser of
the two values by Y, use the cdf of X to write down the
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION
A good model for the variation, from item to item, of a quality
characteristic of a certain manufactured product is a random
variable X with probability density function
$f(x)=\fraccfw_2x
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION A continuous random variable X is uniformally distributed
in the
interval $-1 \leq x \leq 1$. Find E(X) and Var(X). The random
variable Y is defined by $Y=X^2$, Use the cdf of X to show t
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION The time to failure of a new type of lightbulb is thought
to
have an exponential distribution.\
If the reliability of this type of lightbulb at 10.5 weeks is 0.9
find the reliability at 1
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION The time to failure,t in hours, of components follows the
density function $f(t)=\alpha ^2t ecfw_-\alpha t, t>0$.\
\begincfw_description
\item[(a)]
What is the probability that a componen
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION
A random variable X has age specific failure rate function
$\lambda (x)=x$. Find f(x) and F(x) and derive the mode and
median of the distribution of x.
ANSWER $G(x)=e^cfw_-\int\lambda(x)\
\documentclass[a4paper,12pt]cfw_article
\begincfw_document
QUESTION Given the continuous pdf $f(x)=\fraccfw_2cfw_x^2,\ 1 \leq x
\leq 2$,
determine the mean and variance of x and find the probability
that x exceeds 1.5. Calculate also the median and the qu
QUESTION
Prove
(a) using the factorial definition,
(b) using the word definition
(i)
(ii)
!
n
nr
=
n
r
!
!
=
n
nr
!
n!
n!
=
=
=
(n r)!(n (n r)!
(n r)!r!
n+1
r
ANSWER
(i) (a)
!
n
r
!
+
!
n
r1
n
r
!
n
(b)
is the number of ways of choosing r from n, each cho
QUESTION The length of time a customer is in a queue waiting to be
served at a certain cash point has cdf F (x) = 1pex , x 0, > 0, 0 < p <
1. Find P(x=0) and the pdf for x>0. Hence find the mean and the variance
of the queueing time.
ANSWER F (x) = 1 pex
QUESTION The length of a certain type of battery is normally distributed with mean 5.0cm and standard deviation .05cm. Find the probability that such a battery has a length between 4.92 and 5.08 cm.
tubes are manufactured to contain 4 such batteries. 95%
QUESTION Given the continuous pdf f (x) = x22 , 1 x 2, determine
the mean and variance of x and find the probability that x exceeds 1.5. Calculate also the median and the quartile range for x and state the interquartile
range (Q3 Q1 ).
ANSWER f (x) = x22
QUESTION
A contractor rents out a piece of heavy equipment for t hours and is paid
50 per hour. The equipment tends to overheat and if it overheats x times
during the hiring period the contractor will have to pay a repair cost x2 .
The number of times the
QUESTION A random sample of size n is taken from a population of size N.
Write down the number of distinct samples when sampling is
(i) ordered, with replacement
(ii) ordered, without replacement
(iii) unordered, without replacement
(iv) unordered, with r
QUESTION
An electronic system receives signals as input and sends out appropriate
coded messages as output.
The system consists of 3 converters C1 , C2 and C3 , 2 monitors M1 and M2
and a perfectly reliable three way switch for connecting the input to the
QUESTION
A and B are two independent events. A is twice as likely to occur as B and
three times as likely to occur as the event that neither A or B does. Find
the probability of A.
ANSWER
A and B are independent therefore P (A B) = P (A)P (B)
A is twice a
QUESTION
You are given the following probabilities relating to two events A and B,
P (A) = 0.5, P (B) = 0.7, P (A or B) = 0.8. Calculate
(i) P (A and B)
(ii) P (A and not B)
(iii) P (A|B)
ANSWER
(i)
P (A and B) = P (A) + P (B)P (A or B) by addition theore
QUESTION A good model for the variation, from item to item, of a
quality characteristic of a certain manufactured product is a random variable
X with probability density function f (x) = 2x2 , 0 x . Each of the
manufactured items is tested and items for w
QUESTION The time to failure of a new type of lightbulb is thought to
have an exponential distribution.
If the reliability of this type of lightbulb at 10.5 weeks is 0.9 find the reliability
at 10 weeks. (Note that reliability is defined as the probabilit