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4 Random Variables and Probability Distributions Online Quiz Questions

Use this free practice quiz with 20 questions to review 4 Random Variables and Probability Distributions, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which classification is appropriate for these two random variables: the number of defective items in a shipment and a person's height?

  1. A

    The number of defective items in a shipment is continuous, and a person's height is discrete.

  2. B

    The number of defective items in a shipment is discrete, and a person's height is continuous.

  3. C

    Both variables are discrete because both are measured in practice.

  4. D

    Both variables are continuous because both describe real-world quantities.

02
Choose one
1 point

Which situation is most naturally modeled by a Bernoulli random variable?

  1. A

    The number of customers arriving in an hour, with any nonnegative integer value.

  2. B

    The waiting time until the next customer arrives, with any nonnegative real value.

  3. C

    Whether one manufactured component is defective, coded as 1 for defective and 0 otherwise.

  4. D

    The number of heads in ten coin tosses.

03
Choose one
1 point

Which statement must be true of the cumulative distribution function FX(x)=P(X≤x)F_X(x)=P(X\le x)?

  1. A

    A CDF is nondecreasing as its argument increases.

  2. B

    A CDF must be strictly increasing at every value.

  3. C

    A CDF assigns probability only to exact individual values.

  4. D

    A CDF can take values greater than 1 when the variable is continuous.

04
Choose one
1 point

Suppose XX is uniformly distributed on [0,20][0,20]. What is P(5≤X≤8)P(5\le X\le 8)?

  1. A

    0.05

  2. B

    0.10

  3. C

    0.20

  4. D

    0.15

05
Choose all
1 point

Select all conditions required for a binomial model.

  1. A

    There is a fixed number of trials.

  2. B

    Each trial has two possible outcomes.

  3. C

    The success probability may change from trial to trial.

  4. D

    The trials are independent, or are treated as approximately independent.

  5. E

    The experiment must measure a continuous variable.

06
Choose all
1 point

Select all statements that correctly describe a continuous random variable and its probability density function.

  1. A

    The PDF must be nonnegative.

  2. B

    The total area under the PDF must be 1.

  3. C

    The height of the PDF at a point is the probability of that exact point.

  4. D

    The probability of any one exact value is 0.

  5. E

    A PDF is used only for count-valued random variables.

07
True or false
1 point

True or false: The expected value of a random variable must be one of the values that the random variable can actually take.

  1. A

    True

  2. B

    False

08
True or false
1 point

True or false: A cumulative distribution function is defined for both discrete and continuous random variables.

  1. A

    True

  2. B

    False

09
Written response
1 point

A geometric random variable counts the trial on which the first success occurs. If the success probability is 0.20.2, what is P(X=3)P(X=3)? Enter the exact decimal value; the absolute tolerance is 0.

10
Written response
1 point

What term describes a numerical function that assigns a number to each outcome of a random experiment?

11
Fill in the blank
1 point

Complete the two conditions for a valid probability mass function: pX(x)p_X(x) must be at least for every possible xx, and ∑xpX(x)\sum_x p_X(x) must equal .

12
Fill in the blank
1 point

For an exponential random variable with rate λ>0\lambda>0, complete the CDF for x≥0x\ge 0: FX(x)=F_X(x)=.

13
Open ended
1 point

Let Y=2X−3Y=2X-3, where E[X]=4E[X]=4 and Var⁡(X)=9\operatorname{Var}(X)=9. Determine E[Y]E[Y] and Var⁡(Y)\operatorname{Var}(Y), and explain which transformation rule is used for each.

14
Choose one
1 point

Which situation is most directly modeled by a Poisson random variable?

  1. A

    The time until the first success in repeated Bernoulli trials with success probability pp.

  2. B

    The number of events in a fixed interval when events occur independently at a constant average rate.

  3. C

    A measurement uniformly distributed over an interval.

  4. D

    The number of successes in a fixed number of independent trials with success probability pp.

15
Choose one
1 point

Which situation is best modeled by a continuous random variable?

  1. A

    The number of customers arriving in an hour is continuous.

  2. B

    Waiting time until the next customer arrives is continuous.

  3. C

    The number of defective items in a shipment is continuous.

  4. D

    The number of heads in ten coin tosses is continuous.

16
True or false
1 point

For a continuous random variable, P(a≤X≤b)=P(a<X<b)P(a\le X\le b)=P(a<X<b).

  1. A

    True

  2. B

    False

17
Choose one
1 point

Let XX be uniformly distributed on [0,20][0,20]. What is P(5≤X≤8)P(5\le X\le 8)?

  1. A

    0.05

  2. B

    0.20

  3. C

    0.15

  4. D

    0.40

18
Choose one
1 point

Which situation satisfies the assumptions of a binomial model?

  1. A

    The number of calls until the first sale when the success probability may change.

  2. B

    The exact weight of ten randomly selected packages.

  3. C

    The number of arrivals in an hour when the average rate is constant.

  4. D

    The number of correct guesses on ten independent true-false questions.

19
Written response
1 point

A random variable has CDF values FX(2)=0.5F_X(2)=0.5 and FX(5)=0.8F_X(5)=0.8. What is P(2<X≤5)P(2<X\le 5)? Enter the probability as a decimal.

20
Written response
1 point

A fair six-sided die has faces numbered 1 through 6. What is its expected value? Enter the expected value as a decimal.