Question 1

If p remains constant in a binomial distribution, an increase in n will increase the variance.

Accepted Answer

The formula for the variance of a binomial distribution is Var(X) = np(1-p). Since p remains constant, an increase in n means an increase in the value of np, which in turn increases the variance.

Question 2

TABLE 5-4
The following table contains the probability distribution for X = the number of traffic accidents reported in a day in Corvallis, Oregon.
 $$\begin{array} { c | c c c c c c } 
\mathrm { X } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline P ( \mathrm { X } ) & 0.10 & 0.20 & 0.45 & 0.15 & 0.05 & 0.05
\end{array}$$
-Referring to Table 5-4, the standard deviation of the number of accidents is ____.

Accepted Answer

The answer of TABLE 5-4
The following table contains the probability...

Question 3

TABLE 5-7
There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution:
 $\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text { Returns }$
$\begin{array}{l}
\begin{array} { c |c c } 
\text { Prob ability}& \text { Neighb or hood A}&\text {  Neighb or hood B } \
\hline .25 & - \$ 22,500 & \$ 30,500 \
.40 & \$ 10,000 & \$ 25,000 \
.35 & \$ 40,500 & \$ 10,500
\end{array}
\end{array}$

-Referring to Table 5-7, if your investment preference is to minimize the amount of risk that you have to take and do not care at all about the expected return, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B?

Accepted Answer

The answer of TABLE 5-7
There are two houses with almost...

Question 4

Another name for the mean of a probability distribution is its expected value.

Accepted Answer

This statement is true. The mean of a probability distribution is calculated by multiplying each possible outcome by its probability and adding up the products. This is the same as finding the expected value of the distribution.

Question 5

If p remains constant in a binomial distribution, an increase in n will not change the mean.

Accepted Answer

The mean of a binomial distribution is given by the formula μ = n * p. If n increases while p remains constant, the mean will increase.

Question 6

If the covariance between two investments is zero, the variance of the sum of the two investments will be equal to the sum of the variances of the investments.

Accepted Answer

When the covariance between two investments is zero, it means there is no linear relationship between their returns. In this case, the formula for the variance of the sum of two investments simplifies to the sum of the variances of the investments. This can be derived mathematically using the following formula:

Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)

When Cov(X,Y) = 0, the formula reduces to:

Var(X + Y) = Var(X) + Var(Y)

Question 7

TABLE 5-8
Two different designs on a new line of winter jackets for the coming winter are available for your manufacturing plants. Your profit (in thousands of dollars) will depend on the taste of the consumers when winter arrives. The probability of the three possible different tastes of the consumers and the corresponding profits are presented in the following table.
 $$\begin{array} { l l c c } 
\hline \text { Probability } & \text { Taste } & \text { Design } A & \text { Design B } \
\hline 0.2 & \text { more conservative } & 180 & 520 \
0.5 & \text { no change } & 230 & 310 \
0.3 & \text { more liberal } & 350 & 270
\end{array}$$

-Referring to Table 5-8, if you decide to choose Design A for 10% of the production lines and Design B for the remaining production lines, what is the expected profit?

Accepted Answer

The answer of TABLE 5-8
Two different designs on a new...

Question 8

The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

Accepted Answer

The Poisson distribution is commonly used to model the number of events that occur in a fixed time interval, such as the number of customers arriving at a department store in a 5-minute period. Therefore, the statement is true.

Question 9

The Department of Commerce in a particular state has determined that the number of small businesses that declare bankruptcy per month is approximately a Poisson distribution with a mean of 6.4. Find the probability that exactly 5 bankruptcies occur next month.

Accepted Answer

The answer of The Department of Commerce in a particular...

Question 10

TABLE 5-3
The following table contains the probability distribution for X = the number of retransmissions necessary to successfully transmit a 1024K data package through a double satellite media.
 $$\begin{array} { l | l l l l } 
X & 0 & 1 & 2 & 3 \
\hline P ( X ) & 0.35 & 0.35 & 0.25 & 0.05
\end{array}$$
-Referring to Table 5-3, the standard deviation of the number of retransmissions is____

Accepted Answer

The answer of TABLE 5-3
The following table contains the probability...

Question 11

In a game called Taxation and Evasion, a player rolls a pair of dice. If on any turn the sum is 7, 11, or 12, the player gets audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The standard deviation of the number of times she will be audited is____ .

Accepted Answer

The answer of In a game called Taxation and Evasion,...

Question 12

The expected return of a two-asset portfolio is equal to the product of the weight assigned to the first asset and the expected return of the first asset plus the product of the weight assigned to the second asset and the expected return of the second asset.

Accepted Answer

This is the formula for calculating the expected return of a two-asset portfolio according to the principles of modern portfolio theory.

Question 13

A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that none of the members chosen for the team have any competition experience?

Accepted Answer

The answer of A debate team of 4 members for...

Question 14

The local police department must write, on average, 5 tickets a day to keep department revenues at budgeted levels. Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.4 tickets per day. Find the probability that less than 6 tickets are written on a randomly selected day from this population.

Accepted Answer

The answer of The local police department must write, on...

Question 15

A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students. Ten of the 15 students have no prior competition experience while the others have some degree of experience. What is the probability that exactly half of the members chosen for the team have some prior competition experience?

Accepted Answer

The answer of A debate team of 4 members for...

Question 16

$$\text { TABLE 5-6 }$$ The quality control manager of Marilyn's Cookies is inspecting a batch of chocolate chip cookies. When the production process is in control, the average number of chocolate chip parts per cookie is 6.0.
-Referring to Table 5-6, what is the probability that any particular cookie being inspected has between 5.0 and 8.0 inclusive chip parts.

Accepted Answer

The answer of $$\text { TABLE 5-6 }$$ The quality...

Question 17

When using a Poisson distribution to approximate a binomial distribution, the standard deviation of the Poisson distribution is np where n is the sample size and p is the probability of an event of interest.

Accepted Answer

The standard deviation of a Poisson distribution is the square root of its mean, which is λ (lambda), not np. In a Poisson approximation to a binomial distribution, λ is equal to np.

Question 18

TABLE 5-2
A certain type of new business succeeds 60% of the time. Suppose that 3 such businesses open (where they do not compete with each other, so it is reasonable to believe that their relative successes would be independent).
-Referring to Table 5-2, the probability that exactly 1 business succeeds is _____.

Accepted Answer

The answer of TABLE 5-2
A certain type of new business...

Question 19

In a Poisson distribution, the mean and standard deviation are equal.

Accepted Answer

The answer of In a Poisson distribution, the mean and...

Question 20

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process. The average number of airplanes arriving per minute is 3. The probability that exactly 6 planes arrive in the next minute is 0.0504.

Accepted Answer

The probability of exactly k arrivals in a Poisson process with rate lambda per unit time is given by the Poisson distribution:P(k arrivals) = (e^-lambda * lambda^k) / k!In this case, lambda = 3 and k = 6. Plugging these values into the formula, we get:P(6 arrivals) = (e^-3 * 3^6) / 6! = 0.0504Therefore, the probability of exactly 6 planes arriving in the next minute is 0.0504, which is the same as the given value.

Quiz 5: Some Important Discrete Probability Distributions