Question 1

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 2

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

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time. The expected number of events in a Poisson distribution is equal to the mean, which in this case is 3 customers per minute.

Therefore, the expected number of customer arrivals for a five-minute period is:

Expected number of customer arrivals = 3 x 5 = 15

So the correct answer is A) 15.

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 variance of X.
A)3
B)9
C)1.5
D)1.7

Accepted Answer

= μ = 3

Question 4

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)0.75
D)0.87

Accepted Answer

The possible values for X in this experiment are 0, 1, 2, or 3. 
The mean or expected value of X is calculated as follows:

E(X) = (0+1+2+3)/4 = 1.5

The variance of X is calculated as follows:

Var(X) = [(0-1.5)² + (1-1.5)² + (2-1.5)² + (3-1.5)²]/4 = 3/4

The standard deviation of X is the square root of the variance:

SD(X) = √(3/4) = 0.87

Therefore, the best choice is D.

Question 5

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

Accepted Answer

To find the expected value of X, we multiply each possible outcome by its probability and then sum them up. So, E(X) = (0.10)(0) + (0.30)(1) + (0.50)(2) + (0.10)(3) = 0 + 0.3 + 1.0 + 0.3 = 1.6. Therefore, the closest option to the expected value calculated is option D, which is 1.8.

Question 6

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 7

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

The expected value of a single die roll is the average value of all possible outcomes. Since a die has six equally likely outcomes (numbers 1 through 6), the expected value is (1+2+3+4+5+6)/6 = 3.5.

Question 8

A pharmaceutical company has determined that if a new cholesterol-reducing drug is manufactured (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 $90,000.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

The answer of A pharmaceutical company has determined that if...

Question 9

X has the following probability distribution P(X):  Compute the variance value of X.
A)1.58
B).955
C).912
D).625

Accepted Answer

To compute the variance of X, we first need to find the expected value of X.

E(X) = Σ(x*P(X)) where the sum is taken over all possible values of X.

E(X) = (-2*0.2) + (0*0.4) + (2*0.4) + (4*0.2) = 1.6

Now, to compute the variance of X, we use the formula:

Var(X) = Σ[(x-E(X))^2 * P(X)] where the sum is taken over all possible values of X.

Var(X) = [(-2-1.6)^2*0.2] + [(0-1.6)^2*0.4] + [(2-1.6)^2*0.4] + [(4-1.6)^2*0.2] = 0.912

Therefore, the answer is choice C with a variance value of 0.912.

Question 10

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).0498

Accepted Answer

The answer of Consider a Poisson distribution with an average...

Question 11

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).1494
C).224
D).3734

Accepted Answer

The Poisson distribution formula is:

P(X = k) = (e^(-λ) * λ^k)/k!

Where:
λ = the average number of arrivals/events in the given time period
k = the number of arrivals/events we're interested in finding the probability of

In this case, we know that λ = 3 (3 customers per minute), and we want to find the probability of 3 customers arriving in a minute, so k = 3.

Substituting these values into the formula, we get:

P(X = 3) = (e^(-3) * 3^3)/3! = 0.224

So the probability of 3 customers arriving in a minute is 0.224, or approximately 0.22. Therefore, the best choice is C.

Question 12

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).0108
D).0118

Accepted Answer

The answer of Consider a Poisson distribution with an average...

Question 13

Historical data for a local steel manufacturing company show that the average number of defects per standard sheet of steel 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 sheets of metal.
A)2
B)32
C)64
D)8

Accepted Answer

The Poisson distribution has the property that the mean and the variance are equal. Therefore, the variance of the number of defects per sheet is also 2. 
The variance of the sum (or average) of independent Poisson distributions is simply the sum (or average) of the variances. Therefore, the variance of the number of defects for 32 sheets is 2*32 = 64. 
The standard deviation is the square root of the variance, which is sqrt(64) = 8. Therefore, the correct choice is D.

Question 14

Historical data for a local steel manufacturing company show that the average number of defects per standard sheet of steel 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

The answer of Historical data for a local steel manufacturing...

Question 15

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 16

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).3370

Accepted Answer

The answer of For a binomial process,the probability of success...

Question 17

Historical data for a local steel manufacturing company show that the average number of defects per standard sheet of steel 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 standard sheets of steel?
A).8750
B).1221
C).0573
D).1396

Accepted Answer

The answer of Historical data for a local steel manufacturing...

Question 18

X has the following probability distribution P(X):  Compute the expected value of X.
A)2.5
B)1.0
C)1.6
D)0.6

Accepted Answer

The answer of X has the following probability distribution P(X):...

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 expected value of X.
A)3
B)9
C)1.5
D)1.7

Accepted Answer

The expected value of a Poisson distribution is equal to its lambda parameter, which in this case is 3. Therefore, the answer is 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 variance for this distribution?
A)1.5
B)1.22
C)0.75
D)0.87

Accepted Answer

The possible outcomes of tossing a fair coin three times are: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.
Since the coin is fair, each outcome has probability of 1/8. The expected value of X is E(X) = 3/2. 
To calculate the variance, we use the formula Var(X) = E(X^2) - [E(X)]^2.
We can find E(X^2) as follows:
E(X^2) = (0^2 + 1^2 + 2^2 + 3^2)/8 = 14/8 = 1.75
Therefore, Var(X) = 1.75 - (3/2)^2 = 1.75 - 2.25 = 0.75. 
Thus, the answer is C.

Quiz 5: Discrete Random Variables

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?

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 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?

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

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.

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

X has the following probability distribution P(X) : Compute the variance value 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 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 probability of 3 customers 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 probability of more than 7 customers arriving within a minute.

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) .

X has the following probability distribution P(X) : Compute the expected value 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 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?