Discrete probability distribution

Use Excel for the following: You will turn in the print of your completed Excel worksheets.
i) First sheet: Discrete probability distribution
Enter your name in cell A1.
Starting in row 10 make columns as follows:
Columncontents
B Y (The numbers 1-9)
C X(i) the values of your UIN (i.e. nine random numbers)
D P(Y) = X(i)/Sum(X(1):X(9))
E cumulative Prob,
F Y*P(Y),
G Y^2 * P(Y),
H Y-E(Y),
I (Y-E(Y))* P(Y),
J (Y-E(Y))^2 * P(Y),
K (Y-E(Y))^3 * P(Y),
Where the sum of any column has meaning, put it at the bottom of the column. E.g., for P it
should equal 1, but nothing should be at the bottom of Y-E(Y).
Also find the median and put it below E(Y).
Is this an appropriate discrete probability distribution? State on the worksheet below the table
how you know it is.
Find the variance by using both methods discussed in class.
ii) Second sheet: How to simulate Bernoulli experiments?
First, we create 100 random numbers,
= 1, ⋯ ,100 from uniform distribution between zero
& one. Then, define any realization equal to or below 0 < p <1, as success and any realization
greater than p as failure. Define the Bernoulli random variable
, which takes value of 1, when
the realization is a “success” and zero, when the realization is “failure”, i.e. = 1 ≤
and = 0 > . It is obvious that the probability of success is p.
Now, do the following in Excel:
I) Create 100 random numbers from uniform distribution between zero & one in cells A1:A100.
II) Create a random variable defined as follows: In cell B1 enter IF(A1< = 0.4, 1, 0). Copy and
paste in the cells B2: B100. This assigns 1 if the outcome of uniform distribution is 0.4 or below
and 0 if the outcome of uniform distribution is above 0.4.
III) In cell C1, find the sum of success (=SUM(B1:B100)) and In cell C2, find the proportion of
success. Is it equal to 0.4?
IV) What happens if you create 1000 random numbers (Do this in cells D1:D1000 and find the
sum and proportion of success in cells F1 and F2).
V) What happens if you create 10,000 random numbers (Do this in cells G1:G10000 and find the
sum and proportion of success in cells I1 and I2).
iii) Third sheet: Binomial distribution
Enter your name in cell A1.
Let p = X(9)*.05 + .25 where X(9) is the ninth digit of your university identity number. This is the
p in the binomial formula.
Now for a sample of size 10, n=10, we will calculate the probability of each event from Y=0 to 10
successes.
Make columns as follows:
Y,
= (
),
, (1 − )

, ( = )), Y*P,
2
∗ , ( − ())
2

Where the sum of any column has meaning, put it at the bottom of the column.
(Label the mean and variance.)
Compute the mean and variance using the binomial formulas from class.
iv) Fourth sheet: Poisson as an approximation of Binomial distribution
Enter your name in cell A1.
Find the probability values for two binomial distributions and contrast them with the Poisson
approximation. You will find two binomial distributions and a Poisson.
For the first binomial: set n = last three digits of your university identification number. If the
value is less than 100, add 100. Then, set p = next preceding digit*.001, but if zero use the prior
value.
For the second binomial: let n be 10 times greater and p be 1/10 of the values of the first
binomial.
The purpose is to observe how closely the Poisson distribution is to the binomial.
Make a table like the following but let Y range up to 20.
Y Binomial Binomial Poisson (These columns look at the difference
between each Binomial column and the
n 339 3390 Poisson column.)
p 0.008 0.0008 Error of Poisson Percent Error
lambda 2.712 339 3390 339 339
0 0.065684 0.066332 0.066404 0.000720 7.203E-05 1.10% 0.11%
1 0.17957 0.180036 0.180087 0.000517 5.133E-05 0.29% 0.03%
2 0.244737 0.244252 0.244198 -0.000539 -5.379E-05 -0.22% -0.02%
3 0.221711 0.22085 0.220755 -0.000955 -9.504E-05 -0.43% -0.04%
At the bottom of the table, type answers to the following:
What happens to the percentage difference as Y increases?
Does this difference matter? Why?
Part II. Solve the following questions:
1) Does the following represents a valid probability table? Find the expected value and the
variance.
Outcomes 1 2 3 4 5
Probability 1/2 1/5 1/10 1/10 1/10
2) Suppose that a student takes a multiple choice test. The test has 10 questions, each of which
has 4 possible answers (only one correct). If the student blindly guesses the answer to each
question, do the questions form a sequence of Bernoulli trials? If so, identify the trial outcomes
and the probability of guessing the correct answer p.
3) Candidate A is running for office in a certain district. Twenty persons are selected at random
from the population of registered voters and asked if they prefer candidate A. Do the responses
form a sequence of Bernoulli trials? If so identify the trial outcomes and the meaning of the
parameter p.
4) Suppose that each person in a population, independently of all others, has a certain disease
with probability p∈ (0,1). For a group of k persons, we will compare two strategies. The first is to
test the k persons individually, so that of course, k tests are required. The second strategy is to
pool the blood samples of the k persons and test the pooled sample first. We assume that the
test is negative if and only if all k persons are free of the disease; in this case, just one test is
required. On the other hand, the test is positive if and only if at least one person has the
disease, in which case we then have to test the persons individually; in this case k+1 tests are
required. Thus, let Y denote the number of tests required for the pooled strategy.
a) What is P(Y=1)? What is P(Y = k+1) ?
b) What is E(Y)? What is V(Y)?
c) Show that in terms of expected value, the pooled strategy is better than the basic strategy if
and only if p < pk where pk = 1- (1/k) ^(1/k). Note that pk  0 as k  infinity. What s the
implication for the choice of strategy?
5) If 25 percent of the balls in a certain box are red, and if 15 balls are selected from the box at
random, with replacement, what is the probability that more than four red balls will be
obtained?
6) Suppose an economist is organizing a survey of American minimum wage workers, and is
interested in understanding how many workers that earn the minimum wage are teenagers.
Suppose further that one out of every four minimum wage workers is a teenager. If the
economist finds 80 minimum wage workers for his survey, what’s the probability that he
interviews exactly 14 teenagers? 35 teenagers? What’s the probability that he gets at least 5
teenagers in his survey?
7) The manager of an industrial plant is planning to buy a machine of either type A or type B. For
each day’s operation the number of repairs X, that the machine A needs is a Poisson random
variable with mean 0.96. The daily cost of operating A is CA = 160 + 40 ∗ X
2
. For machine B, let Y
be the random variable indicating the number of daily repairs, which has mean 1.12, and the
daily cost of operating B is CB = 128 + 40 ∗ Y
2
. Assume that the repairs take negligible time and
each night the machine are cleaned so that they operate like new machine at the start of each
day. Which machine minimizes the expected daily cost?
8) The number of calls coming per minute into a hotels reservation center is Poisson random
variable with mean 3.
(a) Find the probability that no calls come in a given 1-minute period.
(b) Assume that the number of calls arriving in two different minutes are independent. Find the
probability that at least two calls will arrive in a given two-minute period.
9) Suppose you have a random variable X whose moments are given by E[Xn
] = n!. Find the
moment generating function for X.
Part III. (optional) In your book, do the following problems:
Chapter 3:
7
th edition: 11, 12, 23, 30, 37, 39, 58, 60, 64, 65, 70, 71, 82, 90, 96, 123, 127, 130, 132, 148, 153,
155, 161
6
th edition: 9, 10, 17, ?? (30 in 7th edition), ?? (37 in 7th edition), 27, 42, 44, 48, 49, 54, 55, 64, 72,
78, ?? (123 in 7th edition), 101, 102, 104, 118, 121, 123, ??(161 in 7th e