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7. Random Variables and Probability Distributions Online Quiz Questions

Use this free practice quiz with 30 questions to review 7. Random Variables and Probability Distributions, test your knowledge, and prepare for your next test or exam.

30 questions
01
Choose one
1 point

Which situation describes a discrete random variable?

  1. A

    The waiting time for a bus, because it can take any value in an interval

  2. B

    The number of customers arriving in an hour, because it has countable possible values

  3. C

    The temperature at noon, because it is measured numerically

  4. D

    The mass of a package, because it can be recorded with decimals

02
Choose one
1 point

Commute times are approximately normal with mean 30 minutes and standard deviation 8 minutes. What is the z-score for a 46-minute commute?

  1. A

    z=−2z=-2

  2. B

    z=0.5z=0.5

  3. C

    z=2z=2

  4. D

    z=5.75z=5.75

03
Choose one
1 point

Which situation is most appropriately modeled by a Poisson distribution?

  1. A

    The number of calls received by a help desk in one hour when calls occur independently at a stable average rate

  2. B

    The number of trials needed to obtain the first success

  3. C

    The number of successes in exactly 20 independent trials with the same success probability

  4. D

    The height of a randomly selected adult

04
Choose all
1 point

Which conditions are required for a binomial random variable? Select all correct choices.

  1. A

    The number of trials is fixed

  2. B

    Each trial has two possible outcomes

  3. C

    The number of trials continues until the first success

  4. D

    The success probability is the same on every trial

  5. E

    The trials are independent

05
Choose all
1 point

Which statements describe a normal distribution? Select all correct choices.

  1. A

    It is symmetric about its mean

  2. B

    Its mean, median, and mode are equal

  3. C

    It has a natural lower bound of zero

  4. D

    Its total area under the curve is 1

  5. E

    Its tails extend indefinitely in both directions

06
True or false
1 point

For a continuous random variable, the probability of taking one exact specified value is 0.

  1. A

    True

  2. B

    False

07
True or false
1 point

If a game has an expected payout of $2, a player must receive exactly $2 on every play.

  1. A

    True

  2. B

    False

08
Written response
1 point

A game pays $10 with probability 0.20 and $0 with probability 0.80. What is the expected payout? Enter the value in dollars.

09
Written response
1 point

Which probability distribution counts the number of independent Bernoulli trials needed to obtain the first success?

10
Fill in the blank
1 point

Complete the statements about a continuous probability density function: The total area under the density curve is , and the probability of one exact value is .

11
Fill in the blank
1 point

For X∼Binomial⁡(n,p)X\sim\operatorname{Binomial}(n,p), complete the formulas: E(X)=E(X)= and Var⁡(X)=\operatorname{Var}(X)=.

12
Open ended
1 point

Explain the difference between discrete and continuous random variables. Include how probabilities are represented for each type and give one example of each.

13
Choose one
1 point

Which statement correctly applies the linearity of expected value to two random variables XX and YY?

  1. A

    E(X+Y)=E(X)E(Y)E(X+Y)=E(X)E(Y) for all random variables

  2. B

    E(X+Y)=E(X)+E(Y)E(X+Y)=E(X)+E(Y) whether or not XX and YY are independent

  3. C

    E(X+Y)=0E(X+Y)=0 whenever the variables are dependent

  4. D

    E(X+Y)=E(X)−E(Y)E(X+Y)=E(X)-E(Y) when the variables are independent

14
Choose all
1 point

When checking whether a probability model is appropriate, which questions should be considered? Select all correct choices.

  1. A

    What random variable is being counted or measured?

  2. B

    Which formula is shortest, regardless of the context?

  3. C

    Is the variable discrete or continuous?

  4. D

    Are the model assumptions, such as independence or a stable rate, reasonable?

  5. E

    Does the model produce a convenient-looking calculation?

15
Choose one
1 point

Which variable is a discrete random variable?

  1. A

    The number of defective products in a sample of 20

  2. B

    The diameter of a manufactured part

  3. C

    The exact temperature of a chemical solution

  4. D

    The waiting time for a bus

16
Choose one
1 point

For a continuous random variable, what does P(X = c) = 0 mean?

  1. A

    The value cannot occur in practice.

  2. B

    The modeled probability at that exact point is zero.

  3. C

    The variable must be discrete.

  4. D

    The total area under the density curve is zero.

17
Choose one
1 point

A game pays $10 with probability 0.20 and $0 with probability 0.80. What is the expected payout per play?

  1. A

    $0.20

  2. B

    $0.80

  3. C

    $2

  4. D

    $10

18
True or false
1 point

True or false: If a game has an expected payout of $2, then every play of the game will pay exactly $2.

  1. A

    True

  2. B

    False

19
True or false
1 point

True or false: Adding 5 minutes to every delivery time changes the mean but leaves the variance unchanged.

  1. A

    True

  2. B

    False

20
True or false
1 point

True or false: If delivery time is measured in minutes, the variance is also measured in minutes.

  1. A

    True

  2. B

    False

21
Written response
1 point

A quality-control sample contains 25 circuit boards, and each board has a 0.04 probability of being defective. Assuming a binomial model, what is the expected number of defective boards?

22
Written response
1 point

Adult heights in a population are approximately normal with mean 72 inches and standard deviation 4 inches. What is the z-score for a height of 68 inches?

23
Written response
1 point

A Bernoulli random variable records success as 1 and failure as 0. If the probability of success is 0.20, what is its variance?

24
Fill in the blank
1 point

A discrete random variable is described by a , which assigns a probability to each possible value.

25
Choose one
1 point

A help desk receives calls according to a Poisson model with an average rate of 4 calls per hour. What is the expected number of calls in one hour?

  1. A

    1 event

  2. B

    4 events

  3. C

    8 events

  4. D

    16 events

26
Choose one
1 point

Which situation is most appropriately modeled by a continuous random variable?

  1. A

    The number of defective products in a sample of 20

  2. B

    The waiting time for a bus

  3. C

    The number of correct answers on a quiz

  4. D

    The number of customers arriving in an hour

27
Choose one
1 point

A discrete random variable X has the following distribution: X = 0, 2, 5 with probabilities 0.50, 0.30, and 0.20, respectively. What is E(X)?

  1. A

    1.60

  2. B

    1.75

  3. C

    2.00

  4. D

    3.50

28
Choose one
1 point

A random variable X has variance 9. What is the variance of X + 5?

  1. A

    4

  2. B

    9

  3. C

    14

  4. D

    25

29
Choose one
1 point

A factory produces circuit boards with a 4% defect rate. A quality-control sample contains 25 boards selected so that defect statuses are approximately independent. Which random variable is appropriately modeled by a binomial distribution?

  1. A

    The number of boards inspected until the first defective board appears

  2. B

    The number of defective boards among 25 approximately independent boards when the defect probability is 0.04

  3. C

    The total weight of 25 circuit boards

  4. D

    The number of defects found while inspecting boards until the factory closes

30
Written response
1 point

A help desk receives events according to a Poisson model with an average rate of 4 events per hour. What is the expected number of events in one hour?