Question 1

Given that X is a discrete random variable, then the laws of expected value and variance can be applied to show that E ( X + 5)= E ( X )+ 5, and V ( X + 5)= V ( X )+ 25.

Accepted Answer

The correct relationship for variance when adding a constant is $V(X + 5) = V(X)$, because adding a constant to a random variable shifts its distribution without affecting its spread.

Question 2

The length of time for which an apartment in a large complex remains vacant is a discrete random variable.

Accepted Answer

The length of time for which an apartment in a large complex remains vacant is considered a continuous random variable because it can take on any value within an interval, including fractions of time units (e.g., days, hours).

Question 3

The binomial random variable is the number of successes that occur in a fixed period of time.

Accepted Answer

The binomial random variable is the number of successes in a fixed number of trials, not a fixed period of time.

Question 4

The number of homeless people in Boston is an example of a discrete random variable.

Accepted Answer

The number of homeless people can only take on certain specific values (such as 0, 1, 2, etc.), making it a discrete random variable.

Question 5

For a random variable X , E ( X + 2)- 5 = E ( X )- 3, where E refers to the expected value.

Accepted Answer

Using linearity of expectation, we have:
E(X + 2) - 5 = E(X) + E(2) - 5 
= E(X) + 2 - 5 
= E(X) - 3 
Therefore, the given equation holds for any random variable X.

Question 6

The amount of milk consumed by a baby in a day is an example of a discrete random variable.

Accepted Answer

The amount of milk consumed by a baby in a day is an example of a continuous random variable because it can take on any value within an interval and is not limited to distinct or separate values.

Question 7

For a random variable X , if V ( cX )= 4 V ( X ), where V refers to the variance, then c must be 2.

Accepted Answer

Using the formula for variance, we have:

V(cX) = E[(cX - E(cX))^2]
      = E[(cX - cE(X))^2]
      = E[c^2(X-E(X))^2]
      = c^2E[(X-E(X))^2]
      = c^2V(X)

Given that V(cX) = 4V(X), we have:

c^2V(X) = 4V(X)

Dividing both sides by V(X), we get:

c^2 = 4

Taking the square root of both sides (since c must be positive), we get:

c = 2

Therefore, the statement "c must be 2" is true.

Question 8

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

Accepted Answer

The binomial distribution is a probability distribution that deals with a fixed number of independent trials, each of which has only two possible outcomes (success or failure).

Question 9

A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

Accepted Answer

This is true, as a continuous variable can have an infinite number of possible values within its range, even if the measurement device used to record it has limitations in precision. For example, a thermometer may only be able to measure temperature to the nearest whole degree Celsius, but the actual temperature could have any value between those whole degrees.

Question 10

The time required to drive from New York to New Mexico is a discrete random variable.

Accepted Answer

The time required to drive from New York to New Mexico is a continuous random variable because it can take on any value within an interval and is not limited to distinct or separate values.

Question 11

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

Accepted Answer

The statement correctly describes the binomial distribution, which is used to model the probability of a certain number of successes in a fixed number of consecutive trials, where each trial has two possible outcomes.

Question 12

A random variable is a function or rule that assigns a number to each outcome of an experiment.

Accepted Answer

A random variable is indeed a function or rule that assigns a number to each outcome of an experiment.

Question 13

The number of home insurance policy holders is an example of a discrete random variable

Accepted Answer

A discrete random variable is a variable that can only take on a finite or countable number of values. The number of home insurance policy holders can only take on a finite number of values (i.e. 0, 1, 2, 3, etc.), making it a discrete random variable.

Question 14

For a random variable X , V ( X + 3)= V ( X + 6), where V refers to the variance.

Accepted Answer

We have

V(X+3) = E[(X+3)^2] - [E(X+3)]^2

= E[X^2 + 6X + 9] - [E(X)+3]^2

= E(X^2) + 6E(X) + 9 - [E(X)^2 + 6E(X) + 9]

= E(X^2) - E(X)^2 = V(X)

Similarly,

V(X+6) = E[(X+6)^2] - [E(X+6)]^2

= E[X^2 + 12X + 36] - [E(X)+6]^2

= E(X^2) + 12E(X) + 36 - [E(X)^2 + 12E(X) + 36]

= E(X^2) - E(X)^2 = V(X)

Hence, V(X+3) = V(X+6) = V(X), which means that Option A is true.

Question 15

A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution.

Accepted Answer

This is a correct definition of a discrete probability distribution.

Question 16

The binomial probability distribution is a discrete probability distribution.

Accepted Answer

The binomial probability distribution only considers discrete variables (such as coin tosses or counts of successes in a fixed number of trials).

Question 17

The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

Accepted Answer

The variance of a binomial distribution is given by the formula np(1-p). Substituting n = 50 and p = 0.20, we get variance = 50*0.20*(1-0.20) = 8. Therefore, the statement is true.

Question 18

The mean of a discrete probability distribution for X is the sum of all possible values of X , divided by the number of possible values of X .

Accepted Answer

The mean of a discrete probability distribution for X is calculated by multiplying each possible value by its probability, then adding up all these values. This result is the weighted average of the possible values, not the sum of all possible values divided by the number of possible values.

Question 19

Faculty rank (professor, associate professor, assistant professor, and lecturer)is an example of a discrete random variable.

Accepted Answer

Faculty rank is a categorical variable because it describes categories or groups rather than numerical values that can be counted or measured.

Question 20

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

Accepted Answer

This is true. The expected value is the long-run average value that can be expected to be obtained from a probability distribution. It is also called the mean or mathematical expectation.

Quiz 6: Random Variables and Discrete Probability Distributions

Given that X is a discrete random variable, then the laws of expected value and variance can be applied to show that E ( X + 5)= E ( X )+ 5, and V ( X + 5)= V ( X )+ 25.

The length of time for which an apartment in a large complex remains vacant is a discrete random variable.

The binomial random variable is the number of successes that occur in a fixed period of time.

The number of homeless people in Boston is an example of a discrete random variable.

For a random variable X , E ( X + 2)- 5 = E ( X )- 3, where E refers to the expected value.

The amount of milk consumed by a baby in a day is an example of a discrete random variable.

For a random variable X , if V ( cX )= 4 V ( X ), where V refers to the variance, then c must be 2.

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

A continuous variable may take on any value within its relevant range even though the measurement device may not be precise enough to record it.

The time required to drive from New York to New Mexico is a discrete random variable.

The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

A random variable is a function or rule that assigns a number to each outcome of an experiment.

The number of home insurance policy holders is an example of a discrete random variable

For a random variable X , V ( X + 3)= V ( X + 6), where V refers to the variance.

A table, formula, or graph that shows all possible values a random variable can assume, together with their associated probabilities, is referred to as discrete probability distribution.

The binomial probability distribution is a discrete probability distribution.

The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

The mean of a discrete probability distribution for X is the sum of all possible values of X , divided by the number of possible values of X .

Faculty rank (professor, associate professor, assistant professor, and lecturer)is an example of a discrete random variable.

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