Question 1

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. Determine the expected number of customer arrivals for a five-minute period.
A) 15
B) 3
C) 243
D) 125

Accepted Answer

μ = (3)(5) = 15

Question 2

Consider a Poisson distribution with an average of 4 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the standard deviation of X.
A) 2
B) 4
C) 16
D) 1.5

Accepted Answer

σ_x == 2.00

Question 3

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of 3 customers arriving within a minute.
A) 1.00
B) .4232
C) .2240
D) .3734

Accepted Answer

P(X = 3) === .224

Question 4

A pharmaceutical company has determined that if a new cholesterol-reducing drug is manufactured and introduced to the market, the following probability distribution will describe the contribution of this drug to their profits during the next six months.  The company management has decided to market this product if the expected contribution to profit for the next six months is more than $1,000,000. Based on the information given above, should the company begin manufacturing the new drug? Explain your answer.
A) Yes, begin manufacturing.
B) No, do not begin manufacturing.

Accepted Answer

μ_x = .2(−$3,000,000) + .5($5,000,000) + .3($2,000,000) = $2,500,000 $2,500,000 > $1,000,000

Question 5

Historical data for a local manufacturing company show that the average number of defects per product produced is 2. In addition, the number of defects per unit is distributed according to a Poisson distribution. What is the probability that there will be a total of 7 defects on four units?
A) .6063
B) .0902
C) .0034
D) .1396

Accepted Answer

P(X = 7) === .1396

Question 6

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of 10 customers or fewer arriving within a minute.
A) .9998
B) .9990
C) .0008
D) .0002

Accepted Answer

P(X ≤ 10) = 1 − P(X ≥ 11) = 1 − .0002 = .9998

Question 7

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the expected value of X.
A) 3
B) 9
C) 1.5
D) 1.7

Accepted Answer

The expected value of X would be the average in the minute, or 3.

Question 8

X has the following probability distribution P(X).  Compute the expected value of X.
A) 2.5
B) 1.0
C) 1.6
D) .6

Accepted Answer

E[X] = (1)(.1) + (2)(.5) + (3)(.2) + (4)(.2) = 2.5

Question 9

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads). What is the variance for this distribution?
A) 1.5
B) 1.22
C) .75
D) .87

Accepted Answer

E[X] == 0+ (1)+ (2)+ (3)= 1.5= + + + = .75

Question 10

Historical data for a local manufacturing company show that the average number of defects per product produced is 2. In addition, the number of defects per unit is distributed according to a Poisson distribution. Determine the standard deviation of the number of defects for 32 units.
A) 2
B) 32
C) 64
D) 8

Accepted Answer

== 8

Question 11

If you were asked to play a game in which you tossed a fair coin three times and were given $2 for every head you threw, how much would you expect to win on average?
A) $3
B) $2
C) $6
D) $9

Accepted Answer

Expected return = 0+ ($2)(1)+ ($2)(2)+ ($2)(3)= 3.0

Question 12

X has the following probability distribution P(X).  Compute the variance value of X.
A) 1.58
B) .955
C) .850
D) .625

Accepted Answer

E[X] = (1)(.1) + (2)(.5) + (3)(.2) + (4)(.2) = 2.5 σ²^x =(.1) +(.5) +(.2) +(.2) = .85

Question 13

One die is thrown. What is the expected value of the number of dots on the top face of the die?
A) 1.0
B) 3.5
C) 4.0
D) 3.0

Accepted Answer

E[X] = 1+ 2+ 3+ 4+ 5+ 6

Question 14

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of more than 7 customers arriving within a minute.
A) .0216
B) .0081
C) .9881
D) .0118

Accepted Answer

P(X ≥ 8) = .0081 + .0027 + .0008 + .0002 = .0118

Question 15

X has the following probability distribution.  Compute the expected value of X.
A) 1.3
B) 1.0
C) 2.4
D) 1.8

Accepted Answer

E[X] = (−2)(.2) + (−1)(.2) + (1)(.2) + (2)(.2) + (9)(.2) = 1.8

Question 16

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads). Determine the expected number of heads.
A) 1.5
B) 1.0
C) 2.0
D) 1.1

Accepted Answer

E[X] == 0+ (1)+ (2)+ (3)= 1.5

Question 17

Historical data for a local manufacturing company show that the average number of defects per product produced is 2. In addition, the number of defects per unit is distributed according to a Poisson distribution. A batch has just been completed. What is the probability that the first three units manufactured in this batch will contain at least a total of 4 defects?
A) .8488
B) .7149
C) .1512
D) .2851

Accepted Answer

P(X ≥ 4) = 1 − [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
= 1 − (.0025 + .0149 + .0446 + .0892) = .8488

Question 18

For a binomial process, the probability of success is 40 percent and the number of trials is 5. Find P(X > 4).
A) .0102
B) .0778
C) .0870
D) .9898

Accepted Answer

P(X = 5) = (.4)⁵ = .0102

Question 19

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the variance of X.
A) 3
B) 9
C) 1.5
D) 1.7

Accepted Answer

= μ = 3

Question 20

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads). What is the standard deviation for this distribution?
A) 1.5
B) 1.22
C) .75
D) .87

Accepted Answer

E[X] == 0+ (1)+ (2)+ (3)= 1.5= + + + = .75== .866

Quiz 6: Discrete Random Variables

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. Determine the expected number of customer arrivals for a five-minute period.

Consider a Poisson distribution with an average of 4 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the standard deviation of X.

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of 3 customers arriving within a minute.

Historical data for a local manufacturing company show that the average number of defects per product produced is 2. In addition, the number of defects per unit is distributed according to a Poisson distribution. What is the probability that there will be a total of 7 defects on four units?

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of 10 customers or fewer arriving within a minute.

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the expected value of X.

X has the following probability distribution P(X) . Compute the expected value of X.

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads) . What is the variance for this distribution?

Historical data for a local manufacturing company show that the average number of defects per product produced is 2. In addition, the number of defects per unit is distributed according to a Poisson distribution. Determine the standard deviation of the number of defects for 32 units.

If you were asked to play a game in which you tossed a fair coin three times and were given $2 for every head you threw, how much would you expect to win on average?

X has the following probability distribution P(X) . Compute the variance value of X.

One die is thrown. What is the expected value of the number of dots on the top face of the die?

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the probability of more than 7 customers arriving within a minute.

X has the following probability distribution. Compute the expected value of X.

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads) . Determine the expected number of heads.

For a binomial process, the probability of success is 40 percent and the number of trials is 5. Find P(X > 4) .

Consider a Poisson distribution with an average of 3 customers per minute at the local grocery store. If X = the number of arrivals per minute, find the variance of X.

Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads) . What is the standard deviation for this distribution?