sta5328_test2_ramin_shamshiri

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Ramin Shamshiri STA5328, Exam2 Page 1

Test2 Solutions Summer 2009Ramin Shamshiri

Question One (35 Points.) Let X = the number of trials needed before the first success is

observed in a Bernoulli experiment. Then the distribution of X is

p(x) = pqx

for x = 0, 1, 2, and p(x) = 0 otherwise, where, q = 1p.

a) Show that the moment generating function (mgf) of X is 1 tm( t ) p / qe . (10 Points.)tXm( t ) E( e ) by definition of mgfs.

0

tx tx x

all x x

m( t ) e p( x ) e pq

by definition of expected values. Thus,

0 1

xt

tx

pm( t ) p qe

qe

using the sum of a geometric series [since 0 < qet < for any |t| < 1].

b) Find the mean of X using its mgf. (10 Points.)

0

X

t

dE( X ) m t

dt

by definition of relations between moments and mgfs

2

000

2 2

0 0

0

2 20

1

0 1

1 1

1

X t

ttt

t t t

t t

t t

d p d u u' v v' u

dt qe dt v v

( qe ) p( qe ) pqe

qe qe

pqe pq q

p pqe


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c) Find the variance of X using its mgf. (15 Points.)

2

2

22 2

0 00

0

2

22

0

0 0 0 0

4 4 40

0

2

1

1 2 1

1

1 1 2 1 1 2 1 1

11 1

t

tt tt

t

t t t t t

t

t

t t t t

t

t

d d d pqe u' v v' uE( X ) m t m' t

dt dt dt vqe

pqe qe qe qe pqe

qe

pqe qe qe qe pqe qe qe qe pq q q q

qqe qe

p q

4 21 2 1q q / p q q / p

Hence, 2

= Var(X) = E(X2)

2= [q(1+q)/p

2][q/p]

2= [q + q

2q

2] / p

2= q / p

2.


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Question Two (30 Points) Daily computer sales records for a company show that on any one

weekday, the number of computers sold (X) has the following distribution:

Number of Sales (x) 0 1 2 any other value

Probabilities: p(x) = P(X = x) 0.7 0.2 0.1 0

a) Show that the above p(x) is in fact the probability (mass) function of a random variable.(10 Points.)

For any function p(x) to be the probability (mass) function of a discrete random variable

both of the following conditions must be satisfied: 0 p(x) < 1 for all < x < and

1all x

p( x ) . Since both of these conditions are satisfied in this case, the above p(x) is in

fact a probability mass function.

b) Suppose we are interested in the distribution of Y = the number of computers sold on twoweekdays. Find the distribution of Y. (15 Points.)

Y = number of computers sold on a 2-day period. Then, the possible values of Y are 0, 1,2, 3 and 4. Let the pair (a, b) denote a = sales on day 1 and b = sales on day 2, in that

order.

Then, assuming independence of sales on the two days we have,

P(Y = 0) = P[(0,0)] = 0.50.5 = 0.25P(Y = 1) = P[(1,0) or (0,1)] = 20.50.3 = 0.30

P(Y = 2) = P[(2,0) or (1,1) or (0,2)] = 0.20.5 + 0.30.3 + 0.50.2 = 0.29

P(Y = 3) = P[(2,1) or (1,2)] = 20.20.3 = 0.12P(Y = 4) = P[(2,2)] = 0.20.2 = 0.04P(Y = any other value) = 0

Hence

x 0 1 2 3 4 a.o.v

p(x) 0.25 0.30 0.29 0.12 0.04 0

c) What did you assume in part (b)? (5 Points.)Assume that sales on the two days are independent


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Question Three (35 Points.) The probability that any vehicle will turn left at a particular

intersection is . The left turn lane at this intersection has room for 3 vehicles. Suppose we want

to find the probability that to fill up the left-turn lane, six cars must arrive at the intersection

while the red light is on.

a) Define the random variable (say X) in this question? (5 Points.)X = Number of cars needed to fill the left turn lane.

b) Write the name of the distribution of X? (5 Points.)Negative binomial distribution

c) Write the parameters and their values of the distribution of X? (5 Points.)r = 3, p = 0.25

d) Give reasons for your answer in b and c. (10 Points.)(i) We have a Bernoulli trial (turn left or not)(ii) Trials are independent (assumed)(iii)

Constant probability of Success = p = .(iv) Number of Successs = r = 3

(v) Random variable X = number of independent Bernoulli trials to observe 3Successs

e) Find P(Left turn lane will be full after 6th car arrives to turn left)(10 Points.)3 3 3

6

5 1 3 5 4 3 36 6 0 0659

2 4 4 2 1 3 4

!p( ) P( X ) .

!

sta5328_test2_ramin_shamshiri

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