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Quiz Questions on Probability Distributions and Statistical Calculations - Prof. Joel A. N, Exams of Data Analysis & Statistical Methods

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

2010/2011

Uploaded on 05/09/2011

amandatech
amandatech 🇺🇸

11 documents

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Download Quiz Questions on Probability Distributions and Statistical Calculations - Prof. Joel A. N and more Exams Data Analysis & Statistical Methods in PDF only on Docsity!

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

  1. Do all work on this test booklet.
  2. Do not disassemble this test booklet.
  3. Start only when instructed to start.
  4. Stop immediately when instructed to stop.
  5. Use a pencil and write legibly.
  6. Closed book.
  7. Calculators permitted.
  8. (8 points)

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

 

  1. (10 points)

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

f y e

Compute Pr[ X + Y >4.5].

2

z

Z

f z ze

   

1

0

k z

z

Z

k

z e

F z e z e

k

 

  1. (12 points)

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

[ ] ( ) 0.12 0.36 0.72 0.64 0.60 0.60 3.

D

d

E D M e e e e e e

d

     

 

2

2 2 3 4 5 6

2

0

0

[ ] ( ) 0.12 0.72 2.16 2.56 3.00 3.60 12.

D

d

E D M e e e e e e

d

     

2

Var D [ ] 12.16  (3.04) 2.

  1. (8 points)

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

f

X | Y

( x | y )=

e

x / y

e

y

y

e

y

y

e

x / y

u = x du = dx

Uf ( X Y , ) XY

UV

X

V

U

Y

V

Vh 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

  1. (15 points)

The joint density on the random variables X and Y is:

x y

XY

f x y e x y y

 

Compute the covariance of X and Y.

2

x y x

X XY

x x

f x f x y dy e dy e

 

  

2 2

2

0

0

[ ] 2 2 2

x x

x

xe e

E X xe dx

 

0 0

( ) ( , ) 2 2 [1 ]

y y

x y y y

Y XY

f y f x y dx e dx e e

   

2 2 2

0

0

[ ] 2 2 2 2

y y y y y y

y

E Y ye ye dy ye e e e

     

0 0 0 0 0 0

2 2 2

0 0 0

2 2 2 2 2 2

0

[ ] ( , ) 2 2

y y y

x y y x

XY

y

y x x y y y

y y y y y y y

E XY xyf x y dxdy xye dxdy ye xe dxdy

ye xe e dy ye ye y e dy

ye e ye e y e ye e

  

   

 

     

      

[ , ] [ ] [ ] [ ] 1 0.

Cov X Y E XY E X E Y

  1. (8 points)

The joint density on the random variables X and Y is:

( 1)

x y

XY

f x y xe x y

 

Compute

E Y [ | X 5].

( 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

 

E Y [ | X ]

x

so

E Y [ | X 5] 0.

  1. (7 points)

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