Question 1. A box has 4 red and 5 blue marbles. As part of an experiment, one of the marbles is drawn at random, but when it is placed back in the box, 3 additional marbles of the same color are put in with it. Now suppose that we draw another marble. Find the probability that the first marble drawn was blue given that the second marble drawn was red.
Question 2. A well known inventory problem is "the newsvendor problem", described as follows: A newsvendor buys papers for 15 cents each and sells them for 25 cents each, and he cannot return unsold papers. Daily demand has the following distribution and each day's demand is independent of the previous day's demand:
|
# of customers
|
23
|
24
|
25
|
26
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27
|
28
|
29
|
30
|
|
Probability
|
.01
|
.04
|
.10
|
.10
|
.25
|
.25
|
.15
|
.10
|
If the newsvendor stocks too many papers, he suffers a loss attributable to the excess supply. If he stocks too few papers, he loses profit because of the excess demand. It
seems reasonable for the newsvendor to stock some number of papers so as to minimize the expected loss. If we let s represent the number of papers stocked, D represent the daily demand, and L(D,s) the newsvendor's loss for a particular stock level s, then the loss is simply:
L(D, s) = 0.10(D-s) if D > s = 0.15(s-D) if D ≤ s
Find the expected loss for s=26, s=27 and s=28 and indicate which stocking level (s=26 or s=27 or s=28) the newsvendor would prefer if he wants to minimize his expected loss.
Question 3. Students have obtained their scores which are normally distributed with mean 80 and variance 225. What is the cutoff point for the top 30% of the students?
Question 4. X ~ BINOMIAL (100,0.2) and Y ~ BINOMIAL (n,0.1). If Z = 3X + 4Y and E(Z)=100:
a) What is n=?
b) What is Var(Z)=?
Question 5. If X1 represents the number of customers visiting store 1 in one hour and X2 represents the number of customers visiting store 2 in the same hour, then the joint distribution can be presented as in the table below. Furthermore, suppose that the random variable
Y=2X1 +X2
X1
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0
|
1
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P2 (X2)
|
|
X2
|
0
|
.2
|
.2
|
.4
|
|
|
1
|
.3
|
.3
|
.6
|
|
P1 (X1)
|
.5
|
.5
|
|
In the above table, the row for p1(x1) and the column for p2(x2) are marginal probabilities for X1 and X2 respectively. All the other values are joint probabilities. For example the joint probability of X1 =1 and X2 = 0 is 0.2.
a) What is the range space RY of this new random variable Y?
b) Give the probability distribution of Y in tabular form
c) What is E(Y)=?
Question 6.
A project consists of 30 critical activities that need to be completed in sequence. The time it takes to complete each activity is independent and follows a Poisson distribution with mean 4 days. What is the probability that this project will be completed in 130 days? (Note: For a Poisson Random Variable T with rate λ, E(T)=Var(T)= λ i.e. its mean is equal to its variance).