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Solutions to quiz v of stat 4105 - theoretical statistics, spring 2011. It includes calculations on probability distributions, moment generating functions, conditional probabilities, and joint probability distributions.
Typology: Exams
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Stat 4105
Theoretical Statistics
Spring 2011 Name_______________________________ Quiz V
Student No. ___________________________
On my honor, in my work on this test, I have complied fully with the Virginia Tech Honor Code.
Instructions
John tosses a fair six sided die 12 times. Let X represent the number of times that a 4 occurs
in those 12 tosses. Mary then tosses the same die 8 times. Let Y represent the number of times that a
2 occurs in those 8 tosses and let Z = X + Y. Compute Pr[ Z =4].
4 16 4 16
Pr[ 4] ( ) ( ) 0.
z n m z
n m
Z p q p q
z
A single pump is to be used sequentially to drain two liquid reserves. Let X represent the time
to drain the first reserve and Y represent the time to drain the second reserve. Assuming:
1
1
x
X
f x e
and
y
Y
Compute Pr[ X + Y >4.5].
2
z
Z
1
0
k z
z
Z
k
At a local pizza restaurant, the hourly demand, D , for a particular type of pizza has the
following probability mass function:
D
d
d
d
f d d
d
d
d
Construct the moment generating function for the distribution on D and use it to obtain E[ D ] and
Var[ D ].
2 3 4 5 6
D
M e e e e e e
2 3 4 5 6
0
0
D
d
E D M e e e e e e
d
2
2 2 3 4 5 6
2
0
0
D
d
E D M e e e e e e
d
2
Var D [ ] 12.16 (3.04) 2.
The joint density on the random variables X and Y is:
f ( x , y ) =
e
− x / y
e
− y
y
0 < x < , 0 < y <
Find the conditional probability density function and expected value of X given that Y = y.
f
Y
( y )=
∫
0
∞
e
− x / y
e
− y
y
dx =− e
− y
∫
0
∞
−
e
− x / y
y
dx =− e
− y
e
− x / y
|
0
∞
=− e
− y
( 0 − 1 )= e
− y
X | Y
( x | y )=
− x / y
− y
− y
− x / y
u = x du = dx
U f ( X Y , ) X Y
V h X Y ( , ) X / Y
2
f f
x y
J x y
g g
x
x y y y
2
2
x y
x
J x y
y y
y
2
2 2
2
UV
u u
f u v
x y uv u
v v
y
v v
The joint density on the random variables X and Y is:
x y
XY
Compute the covariance of X and Y.
2
x y x
X XY
x x
2 2
2
0
0
x x
x
0 0
y y
x y y y
Y XY
2 2 2
0
0
y y y y y y
0 0 0 0 0 0
2 2 2
0 0 0
2 2 2 2 2 2
0
y y y
x y y x
XY
y
y x x y y y
y y y y y y y
The joint density on the random variables X and Y is:
( 1)
x y
XY
Compute
( 1)
0 0 0
x y x xy x xy x
X
f x xe dy xe e dy e e e
( 1)
|
x y
xy XY
Y X x
X
f x y xe
f y x xe
f x e
x
so
The random variable T has an exponential distribution with
Compute
Pr[ T 45 | T 20].
(0.02)(45)
(0.02)(25)
(0.02)(20)
Pr[ 45 | 20] 0.
T
T
F e
T T e
F e