Free Practice Quiz Question List

04 Random Variables and Probability Distributions Online Quiz Questions

Use this free practice quiz with 20 questions to review 04 Random Variables and Probability Distributions, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Which quantity is a discrete random variable?

A
The exact temperature of a chemical solution
B
The number of customers arriving at a store in one hour
C
The exact time until the next customer arrives
D
The weight of a randomly selected package

True or false

1 point

For a continuous random variable, the probability of observing one exact value, such as $X=5$ , is zero.

A
True
B
False

Written response

1 point

What is the standard name for the function that gives $P(X=x)$ for a discrete random variable?

Choose one

1 point

Which set of probabilities defines a valid discrete probability distribution for $X$ taking the values $0,1,2,3$ ?

A
$P(X=0)=0.25, P(X=1)=0.35, P(X=2)=0.20, P(X=3)=0.30$
B
$P(X=0)=-0.10, P(X=1)=0.40, P(X=2)=0.30, P(X=3)=0.40$
C
$P(X=0)=0.25, P(X=1)=0.35, P(X=2)=0.20, P(X=3)=0.20$
D
$P(X=0)=0.25, P(X=1)=0.35, P(X=2)=0.20, P(X=3)=0.25$

Fill in the blank

1 point

Complete the transformation rule: $\operatorname{Var}(X+c)=$ .

True or false

1 point

The expected value of a discrete random variable must always be one of the values that the random variable can take.

A
True
B
False

Choose all

1 point

Which conditions are required for a binomial distribution? Select all that apply.

A
The number of trials is fixed in advance.
B
Each trial can have several unordered outcome categories.
C
The trials are independent and each has the same probability of success.
D
The number of trials continues until the first success occurs.

Written response

1 point

A random variable $X$ has probabilities $0.20, 0.35, 0.30, 0.15$ for $x=0,1,2,3$ , respectively. What is $E(X)$ ? Enter the decimal value; the answer is in messages per hour.

Choose one

1 point

A waiting time is uniformly distributed on $[0,10]$ minutes. What is the probability that the waiting time is between 2 and 6 minutes?

A
0.20
B
0.40
C
0.60
D
0.80

Fill in the blank

1 point

Complete the CDF rule for a continuous random variable: $P(a<X<b)=$ .

Choose all

1 point

Which statements correctly describe variance or standard deviation? Select all that apply.

A
If time is measured in minutes, variance is measured in minutes squared.
B
Standard deviation is obtained by squaring the variance.
C
A standard deviation of zero means the variable has the greatest possible variability.
D
Standard deviation has the same units as the original random variable.

Choose one

1 point

A temperature variable $C$ has standard deviation $4$ degrees Celsius. If $F=1.8C+32$ , what is the standard deviation of $F$ ?

A
4.0
B
3.2
C
32.0
D
7.2

Open ended

1 point

A discrete random variable has values $0,1,2,3$ with probabilities $0.20,0.35,0.30,0.15$ , respectively. Explain how to calculate its variance and determine the numerical variance, including appropriate units.

Choose one

1 point

Which conditions must a function satisfy to be a valid probability density function for a continuous random variable?

A
$f(x)\geq 0$ for all $x$ , and $\int_{-\infty}^{\infty}f(x)\,dx=1$
B
$f(x)\leq 1$ for all $x$ , and $\int_{-\infty}^{\\infty}f(x)\,dx=0$
C
$f(x)$ must be constant, and its total area may be greater than one
D
$f(x)$ may be negative when the total area is one

Choose one

1 point

Which quantity is best modeled as a continuous random variable?

A
The number of books in a backpack
B
The exact distance driven by a car
C
The number of customers arriving in an hour
D
The number of defective items in a shipment

True or false

1 point

For a continuous random variable, $P(a<X<b)=P(a\leq X\leq b)$ .

A
True
B
False

Choose one

1 point

A discrete random variable has probabilities $0.20$ , $0.35$ , $0.30$ , and $0.15$ for its four possible values. Which conclusion is correct?

A
The distribution is invalid because it contains four possible values.
B
The distribution is invalid because the probabilities must be equal.
C
The distribution is invalid because one probability is greater than one.
D
The distribution is valid because all probabilities are between zero and one and sum to one.

Written response

1 point

A passenger's waiting time $X$ is uniformly distributed from 0 to 10 minutes. What is $E(X)$ , the expected waiting time?

Choose one

1 point

A waiting time is uniformly distributed on $[0,10]$ minutes. What is the probability that the waiting time is between 2 and 6 minutes?

A
0.20
B
0.40
C
0.60
D
0.80

Written response

1 point

For the message distribution with probabilities $0.20, 0.35, 0.30, 0.15$ for $X=0,1,2,3$ , respectively, the variance is $0.94$ . What is the standard deviation, in messages per hour? The nonzero absolute tolerance is 0.01.