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STT441-Spring2011-Midterm2.5 NAME ID
SOLUTION
The following relates to problems 1- 3. A class contains 10 girls and 20 boys. A committee of 5 is to be selected at random from the class.
Problem 1. Let X represents the number of girls in the committee when no student can be selected more than once. How is X distributed? (a) Hypergeometric( 30, 10, 5) (b) Geometric (.5) (c) Hypergeometric( 30, 10, 1) (d) Binomial(5, 2/3) (e) Binomial(5, 1/3)
Problem 2. Let Y represents the number of girls in the committee when students can be selected more than once. How is Y distributed? (a) Hypergeometric( 30, 10, 5) (b) Geometric (.5) (c) Hypergeometric( 30, 10, 1) (d) Binomial(5, 2/3) (e) Binomial(5, 1/3)
Problem 3. It was decided to increase the size of the committee to 30 (observe: 30 is also the size of the class). Find ) )). Hint: What can you say about in this case?
(1) (6.6, 6.7) (2) (0, 6.7) (3) (6, 6.3) (4) (0, 1.4) (5) None of the
above
Solution: ) so ). Also,
) ) (^) ( ) ( )
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The following relates to problems 4-7. The average number of calls that Jill gets in an hour is 2. Problem 4. Let X denotes the number of calls that Jill will get in next hour. Find P ( X 1 ).
Solution: X ~ Poisson ( 2 )
^2
e^
P X P X
Problem 5. Let T be the time( in hours) between now and the next call that Jill will get. Find P ( T 2 ).
(1) e ^2 (2) e ^1 /^2 (3) e ^4 (4) e ^2.^5 (5) e ^5
T~exponential( 2 ) so P ( T 2 ) e ^2 *^2 e ^4
Problem 6. Let U be the time ( in hours) between now and the 3 rd^ call that Jill will get. How is U distributed?
(a) exponential( 2 ) (b) Weibull( 3 , . 4 )
(c) normal( 5 , ^2 1 ) (d) Gamma( 3 , 2 )
(e) Gamma( 2 , 2 )
Problem 7. Find P ( U 1 ). (Hint: { } { })
^2 ^2
e^
PU PX PX P X PX
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The following relates to problems 8-10. Let ( X , Y ) be uniformly distributed on the triangle generated by the points (0,0), (0,2) and (2, 2). More formally the joint density is given by f ( x , y ) 1 / 2 , 0 x y 2.
Problem 8. Complete: f X ( x )= ______ if 0 x 2
a. 4 x b. 2 y c. ( 1 / 2 ) x^2 d. 3 ( 1 x ) e. ( ) )
2 f x dy x y x
X ^
if 0 x 2
Problem 9. Find P ( Y 1 ) (Hint: Look at all ( x , y )from 0 x y 2 where y 1 )
P ( Y 1 )( 1 / 2 )* area ( B )( 1 / 2 )*( 1 / 2 ) 1 / 4
, where B is the triangle generated by the points (0,0), (0,1) and (1, 1).
Problem 10. Complete: f Y | X x ( y )= ______ if x y 2 for each 0 x 2.
x
(2) 3 x (3)
2 y
2 x
(5) x
f x x
f x y f y X
Y X x 2
(1/2)(2-x)
Problem 15. Find E ( X^2 )
(1)3.67 (2) 2.71 (3) 4.67 (4) 5.49 (5) 2.
E ( X^2 )
4 2
(^234) 0
4
4
2
2
2
0
x x dx x x dx x x x
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The following relates to problems 16-18. Let (X, Y, Z) be the number of customers that will end up filling their car with fuel that is(regular, premium, premium+) among the next 5 customers. It is known that P(regular)=.25, P(premium)= .45 and P(premium +)=.3.
Problem 16. Find P(X=1, Y=3, Z=1) (1) .66 (2) .18 (3) .25 (4) .14 (5).
Solution. (X, Y, Z) ~Multinomial(n=5; .25, .45, .3)
P(X=1, Y=3, Z=1)=. 25. 45. 3. 14
Problem 17. How is X distributed? (a) Hypergeometric( 5, 1, 1) (b) Geometric (.25) (c) Poisson( .25) (d) Binomial(5, .75) (e) Binomial(5, .25)
Problem 18. How is distributed? (Hint: Those are the customers who are not using regular)
(a) Hypergeometric( 5, 1, 1) (b) Geometric (.25) (c) Poisson( .25) (d) Binomial(5, .75) (e) Binomial(5, .25)
The following relates to problem 19-20. Let the hazard rate for a random variable be given by )
Problem 19. Find )
(1) (2) (3) (4) (5)
Problem 20. Find the density (pdf) of
(1) (2) (3)
