Please show your work for the following: 4. Define the random variable Y as the
ID: 3125039 • Letter: P
Question
Please show your work for the following:
4. Define the random variable Y as the outcome of a fair die roll times 3 (e.g.,
if a ve is rolled, Y takes a value of 5*3=15). What is the expected value
of Y? What is the variance of Y?
5. A random variable X has E(X)=3 and V(X)=7. Define a new random
variable: Y=6X + 3. What is E(Y) and V(Y)?
6. In class we often talked about a 6-sided die. Now consider a 24-sided die,
with sides numbered 1 to 24 (so that the sample space of outcomes is
f1,2,. . . ,24g). Suppose that all outcomes are equally likely. What is the
mean and variance of the outcome of a roll of a 24-sided die?
7. A firm's revenue distribution for the coming year is (in million dollars):
15% probability of 5, 25% probability of 10, 20% probability of 12, 30%
probability of 14 and 10% probability of 20. If F(.) is the cumulative
distribution function of this random variable, what is F(11) equal to?
What is F(14)? What is the variance of the rm's revenue distribution?
8. Airtran's flight #307 can accomodate 50 passengers, but the flight is over-
booked, as 52 tickets were sold. Each ticketed passenger can arrive late
and miss the flight with a probability 0.02. What is the probability
that no passenger arrives late? What is the probability that exactly one
passanger arrives late? What is the probability that Airtran has to pay
overbooking fees (and reschedule passengers to different
flights)?
9. The world incidence of (frequency of world population having) diabetes is
5%. If 20 persons are chosen at random, what is the probability that no
more than 3 have the disease?
10. Plot in Excel the probability mass function (density) for the following:
a Binomial random variable X with n = 10, p = 0:3
a Binomial random variable Y with n = 100; p = 0:4
a Uniform random variable X with n = 50
Attach the clearly labeled graphs.
Explanation / Answer
Multiple questions. First 3 questions answered.
4. Define the random variable Y as the outcome of a fair die roll times 3 (e.g.,
if a five is rolled, Y takes a value of 5*3=15). What is the expected value
of Y? What is the variance of Y?
Number
Y
P(Y)
y*p(y)
(y-mean)^2*p(y)
1
5
0.1667
0.8335
26.06146
2
10
0.1667
1.667
9.385629
3
15
0.1667
2.5005
1.044794
4
20
0.1667
3.334
1.03896
5
25
0.1667
4.1675
9.368125
6
30
0.1667
5.001
26.03229
Total
1.000
17.5035
72.93126
expected value of Y =17.5035
variance = 72.9313
5. A random variable X has E(X)=3 and V(X)=7. Define a new random
variable: Y=6X + 3. What is E(Y) and V(Y)?
E(aX + b) = aE(X) + b
var(aX + b) = a2 var(X)
E(Y) = 6*3+3 =21
V(Y) = 62*7=252
6. In class we often talked about a 6-sided die. Now consider a 24-sided die,
with sides numbered 1 to 24 (so that the sample space of outcomes is
f1,2,. . . ,24g). Suppose that all outcomes are equally likely. What is the
mean and variance of the outcome of a roll of a 24-sided die?
X
P(X)
x*p(x)
(x-mean)^2*p(x)
1
0.04167
0.04167
11.34947
2
0.04167
0.08334
10.01574
3
0.04167
0.12501
8.765348
4
0.04167
0.16668
7.598296
5
0.04167
0.20835
6.514584
6
0.04167
0.25002
5.514212
7
0.04167
0.29169
4.597181
8
0.04167
0.33336
3.763489
9
0.04167
0.37503
3.013137
10
0.04167
0.4167
2.346126
11
0.04167
0.45837
1.762454
12
0.04167
0.50004
1.262122
13
0.04167
0.54171
0.845131
14
0.04167
0.58338
0.511479
15
0.04167
0.62505
0.261167
16
0.04167
0.66672
0.094196
17
0.04167
0.70839
0.010564
18
0.04167
0.75006
0.010272
19
0.04167
0.79173
0.09332
20
0.04167
0.8334
0.259709
21
0.04167
0.87507
0.509437
22
0.04167
0.91674
0.842505
23
0.04167
0.95841
1.258914
24
0.04167
1.00008
1.758662
1.000
12.501
72.95752
Mean =12.501
Variance =72.9575
Number
Y
P(Y)
y*p(y)
(y-mean)^2*p(y)
1
5
0.1667
0.8335
26.06146
2
10
0.1667
1.667
9.385629
3
15
0.1667
2.5005
1.044794
4
20
0.1667
3.334
1.03896
5
25
0.1667
4.1675
9.368125
6
30
0.1667
5.001
26.03229
Total
1.000
17.5035
72.93126