Homework 4 Solutions
Discrete Probability Distribution
1.
Consider the following discrete probability distribution
=
=
=
=
=
=
otherwise
x
x
x
x
x
X
P
0
16
,
4
.
0
8
,
25
.
0
4
,
15
.
0
2
,
2
.
0
)
(
Compute
a.
E[X] = 0.2*2+0.15*4+0.25*8+0.4*16=9.4
b.
E[2X+3]=2*E[X]+3=15.8
c.
Var[X] = 33.24
Standard deviation of X= sqrt(Var(X))=5.766
d.
Var[2X+3]=4*Var[X]=132.96
2.
Let X be a random variable with distribution function P(X ≤ x) = F(x) and let
Y = 1+3X be another random variable. If E[Y] = 10 and Var[Y]=16.
Compute the
E[X] and Var[X].
Solution:
E[Y]=1+3E[X]=10 therefore E[X]=3, Var[Y]=9Var[X]=16 therefore
Var[X]=16/9.
3.
The price of a stock in a given trading day
changes
according to the following
distribution
=
=
=
−
=
=
=
2
125
.
0
1
125
.
0
0
5
.
0
1
25
.
0
)
(
x
x
x
x
x
X
P
Find the distribution of the
change
in stock price after two trading days assuming that
the changes in two consecutive days are independent. Find the expectation and the
variance of the total change in two days.
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Solution:
Let the random variable X1 represent the change in the first day and X2
change in the second day. So the total change in two consecutive days will be X1+X2.
The distribution of this random variable is
=
=
=
=
=
−
=
−
=
=
=
+
4
3
64
/
1
64
/
2
2
64
/
9
1
16
/
3
0
16
/
5
1
16
/
4
2
16
/
1
)
2
1
(
x
x
x
x
x
x
x
x
X
X
P
Note that E[X] = 0.125 and Var[X] = 0.8594
E[X1+X2] = E[X1] + E[X2] = 2E[X] = 2*0.125=0.25
Var[X1+X2] = Var[X1] + Var [X2] = 2Var[X] = 1.719
4.
A die is rolled twice. Let X denotes the sum of two numbers that turn up (specifically
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 Winter '07
 All
 var, Discrete probability distribution, Briarwood Mall

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