ISEN 609
Quiz 3
Oral Quiz
Oral Quiz
There are 25 questions in total
are 25 questions in total
At the end of the quiz, based on your
answers score on 10 would be assigned
answers a score on 10 would be assigned
We will randomly select the person who get

ISEN 609: Probability for Engineering Decisions
SUMMER 2011
340A Zachry, MTWRF 12:00-1:35 pm
Instructor
N. Gautam, 235A Zachry, 845-5458, gautam@tamu.edu
Oce Hours Tuesdays and Thursdays 2:30-4:30pm
TA
Ang Li, 303D Zachry, isela@tamu.edu
Prerequisites An

1
Laplace Stieltjes Transforms
Consider a non-negative-valued continuous random variable X . The Laplace Stieltjes Transform (LST)
of X is given by
FX (s) = E [esX ].
Therefore mathematically the LST can be written (and computed) as
FX (s) =
esx dFX (x) =

Discrete Probability Distributions
NOTE : This document could contain errors, please be warned. The text could possibly use other
notation and format. Also note that the PMF is P cfw_X = x, we call it p(x) but others call it f (x).
1. Discrete Uniform Dis

Class Problems on Exponential Distribution and Poisson Processes
1. The amount of time it takes to spot a defect in a cast is distributed exponentially with mean 15
minutes.
a. What is the probability of spotting a defect within 10 minutes?
b. If I have b

Multivariate (actually mostly bivariate!) Distributions
DISCRETE: Let X and Y be discrete random variables. Then the joint pmf of X and Y is
pX,Y (x, y ) = P (X = x, Y = y ).
[Note: Read P (X = x, Y = y ) as P (X = x and Y = y ).]
An important property th

Class Problems: DTMC analysis
1. A company uses two forecasting tools to predict the demand of its product. Tool i is eective w.p.
pi (for i = 1, 2). If the nth prediction uses tool i and it is observed to be eective, then the (n + 1)st
prediction is also

CTMC Modeling Exercises
1. Consider a machine that operates for an exp() amount of time and then fails. Once it
fails, it gets repaired. The repair time is an exp() random variable, and is independent
of the past. The machine is as good as new after the r

Class Problems: Computing Expectations (discrete)
1. Compute the expected value of a binomial random variable with paramaters n and p
by writing it down as a sum of n Bernoulli random variables.
2. The following problem was posed and solved in the 18th ce

Class Problems on Bayes Rule and Discrete Random Variables
1. In a city 1% of the population has cancer. A cancer test is eective 98% of the time. If a
person has been tested positive for cancer, what is the probability that he/she actually has
cancer?
2.

Class Problems: DTMC
1. A company uses two forecasting tools to predict the demand of its product. Tool i is eective
w.p. pi (for i = 1, 2). If the nth prediction uses tool i and it is observed to be eective, then
the (n + 1)st prediction is also done usi

Untitled.notebook
August13,2015
1
Solution to Class Problem 3, second part
Let Z be the time of next arrival, Z exp() with = 1/10 per minute. Recall that Y2 and Y3
are remaining service for barbers 2 and 3 respectively, with Y2 exp(2 ) and Y3 exp(3 ) wher

Extra problems on Topic 1 in ISEN 609
1. There are two mutually independent paths to go from location A to location B. The rst path
takes a random time exponentially distributed with mean 10 minutes, while the second path
takes a random time uniformly dis

Homework 3
1. Suppose the pdf of X is given by
f (x) =
1
xex/2 ,
4
0,
x>0
otherwise.
Calculate the LST. Using the LST calculate E(X) and also verify E(X) using the pdf
directly.
2. Bus A will arrive at a station at a random time uniformly distributed betw

The newsvendor problem : a probabilistic EOQ model
Consider a newsstand (or a newspaper vending machine) that stocks S newspapers
early in the morning, everyday. The daily demand for newspaper is a random variable with PDF P (x), where P (x) is the probab

Solutions to homework 3
1
2s + 1
1. The LST, F (s) =
2
. Also, E(X) = - F (0) = 4.
2. Let X and Y denote the number of minutes past 10:00 a.m. that bus A and bus B arrive at the station,
respectively. X is uniformly distributed over (0, 30). Given that X

Class Problems
1. The following problem was posed and solved in the 18th century by Daniel Bernoulli.
Suppose that a jar contains 2N cards, two of them marked 1, two marked 2, two
marked 3, and so on. Draw out m cards at random. What is the expected numbe

1
Laplace Stieltjes Transforms
Consider a non-negative-valued continuous random variable X. The Laplace Stieltjes Transform (LST)
of X is given by
FX (s) = E[esX ].
Therefore mathematically the LST can be written (and computed) as
FX (s) =
esx dFX (x) =
e

Homework 2
1. Let the probability density function of X be given by
fX (x) =
c(4x 2x2 ), 0 < x < 2
0,
otherwise.
(a) What is the value of c?
(b) What is the cumulative distribution function of X?
1
(c) P cfw_ 2 < X < 3 = ?
2
2. The median of a continuous

Solutions to homework 2
1. Using the properties of cdfs
(a) Solving for c in
f (x)dx = 1, we get c = 3/8.
(b) We have
x0
0,
3
(2x2 2 x3 ), 0 x 2
F (x) =
3
8
1,
x 2.
(c) Since
Pcfw_
1
3
1
3
< X < + P cfw_X = P cfw_X
2
2
2
2
1
We have P cfw_ 2 < X < 3

Discrete Probability Distributions
NOTE : This document could contain errors, please be warned. The text could possibly use other
notation and format. Also note that the PMF is P cfw_X = x, we call it p(x) but others call it f (x).
1. Discrete Uniform Dis

Solutions to homework 1
1. Let Ai denote the event that a 5 occurs on the ith roll and Bi denote the event that a 7 occurs on the
ith roll. Then En = [n1 (Ai Bi )c ] An . Since each roll of the dice is independent and Ai and Bi
i=1
are disjoint,
n1
P (En

Homework 1
1. A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability
that a 5 occurs rst. (You are welcome to verify the solution using a simple
simulation)
Hint: Let En denote the event that a 5 occurs on the nth roll and no