Download E(X)

STT 315, Section 201
Worksheet 3
07/14/2014
SOLUTIONS
1.
X
P(X=x)
E(X)= 1.2
X
P(X=x)
0
0.2
V(X)=0.56
1
0.4
2
0.4
S.D(X)= 0.7483
100
0.1
200
0.2
300
0.5
400
0.2
E(X)=280 V(X)=7600 S.D(X)=87.178
2. A day trader buys an option on a stock that will return a $100 profit if the stock goes up today
and loses $200 if the stock goes down. If the trader thinks that there is a 75% chance that the
stock will go up.
a) Create a probability model
X
100
-200
P(X)
0.75
0.25
b) What is the expected value of the option and its standard deviation?
E(X)= 100*0.75-200*0.25=0.25 SD(X)=129.904
3. You draw a card from a deck of cards. If you get a red card, you win nothing. If you get a
spade you win $5. For any club you win $10 plus an extra $20 for the ace of clubs.
a) Create a probability model:
X
0
5
10
P(X)
1/2
1/4
12/52
b) Find the expected amount you will win at this game?
E(X)=0.5*0+0.25*5+(12/52)*10+(1/52)*30=$4.135
30
1/52
4. You play two games with the same opponent. The probability you win the first game is 0.4. If
you win the first game, the probability that you also win the second game is 0.2. if you lose the
first game, the probability that you the second is 0.3.
a)
b)
c)
d)
Are the two games independent? No
What’s the probability that you lose both games? =0.6*0.7=0.42
What’s the probability that you win both games? =0.4*0.2=0.08
Let random variable X denote the number of games you win. Create a model for X.
X
0
1
2
P(X)
0.42
0.5
0.08
STT 315, Section 201
Worksheet 3
07/14/2014
e) What are the expected value and standard deviation? E(X)=0.66
SD(X)=0.62
5. Random Variables: Given two independent random variables with means and standard
deviations as shown below:
Mean
10
20
X
Y
S.D
2
5
Find the mean and standard deviation for the following:
Linear Combination of RVS
a) 3X
b) Y+7
c) -0.5Y
d) X-Y
e) X+Y
f) 4X-10Y
Mean
=3*E(X)=10
=E(Y)+7=27
=-0.5*E(Y)=-0.5*20=-10
=E(X)-E(Y)=-10
=E(X)+E(Y)=30
=4*E(X)-10*E(Y)=-160
Standard Deviation
=√ 9*V(X)=√ 36=6
=√ V(Y)=√ 25=5
=√ (-0.5)2*V(Y)=√0.25*25=5
=√ V(X)+V(Y)=√ (25+4)=√29
=√ V(X)+V(Y)=√ (25+4)=√29
=√ 16*V(X)+100V(Y)=√
2564=50.64
6. A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.6 with a
standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them.
a) How many broken eggs do you expect to get?E(3*X)=3*E(X)=1.8
b) What’s the standard deviation? SD(3*X)=√ 9*V(X)=√ 2.25=1.5
c) What assumption did you have to make about the eggs in order to answer the above
questions? Each set of a dozen eggs is independent of the other
7. Suppose the time is takes a customer to get and pay for seats at the ticket window of a baseball
park is a random variable with a mean of 100 seconds and a standard deviation of 50 seconds.
When you get there you only find two people in line in front of you.
a) How long do you expect to wait for your turn to get tickets? E(2*X)=200
b) What’s the standard deviation of your wait time? SD(2*X)=√ 4*V(X)=100
c) What assumption are you making about the two customers in finding the standard
deviation? The people standing in line have independent waiting times to
purchase the tickets.