1A single experiment with only two possible outcomes, 'success' or 'failure', is known as a...
bernoulli process
Easy
A.Geometric trial
B.Poisson trial
C.Binomial trial
D.Bernoulli trial
Correct Answer: Bernoulli trial
Explanation:
A Bernoulli trial is the fundamental building block of many discrete distributions and is defined as a single experiment with two mutually exclusive outcomes.
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2The Binomial distribution is used to model the number of...
binomial distribution
Easy
A.trials until the first success.
B.trials until a fixed number of successes.
C.successes in a fixed number of trials.
D.events in a fixed interval of time or space.
Correct Answer: successes in a fixed number of trials.
Explanation:
The Binomial distribution finds the probability of getting a specific number of successes (e.g., successes) in a fixed number of independent Bernoulli trials (e.g., trials).
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3Which distribution is used to model the number of trials needed to get the first success?
geometric distribution
Easy
A.Binomial distribution
B.Poisson distribution
C.Geometric distribution
D.Bernoulli distribution
Correct Answer: Geometric distribution
Explanation:
The Geometric distribution specifically models the waiting time, in terms of the number of trials, for the very first success in a sequence of independent Bernoulli trials.
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4Which of the following scenarios is best modeled by a Poisson distribution?
poisson distribution
Easy
A.The number of heads in 10 coin flips.
B.The number of emails received by an office in one hour.
C.The height of students in a class.
D.The number of cards drawn from a deck until an ace is found.
Correct Answer: The number of emails received by an office in one hour.
Explanation:
The Poisson distribution is ideal for modeling the number of events that occur within a fixed interval of time or space, when the events happen independently and at a constant average rate.
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5The Negative Binomial distribution is a generalization of which other distribution?
negative binomial distribution
Easy
A.Poisson distribution
B.Normal distribution
C.Geometric distribution
D.Uniform distribution
Correct Answer: Geometric distribution
Explanation:
The Geometric distribution models the number of trials until the 1st success. The Negative Binomial distribution generalizes this to model the number of trials until the -th success, where .
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6The Moment Generating Function (MGF) for a Bernoulli distribution with probability of success is given by:
moment generating function ( without proof) of the above mentioned distributions
Easy
A., where
B.
C.
D.
Correct Answer: , where
Explanation:
The MGF for a Bernoulli random variable is . Since can be 0 (with probability ) or 1 (with probability ), the MGF is .
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7What are the two parameters that define a Binomial distribution?
binomial distribution
Easy
A.The probability of success () and the probability of failure ().
B.The rate parameter () and the number of successes ().
C.The mean () and the variance ().
D.The number of trials () and the probability of success ().
Correct Answer: The number of trials () and the probability of success ().
Explanation:
A Binomial distribution, denoted as , is fully defined by two parameters: , the total number of independent trials, and , the probability of success on a single trial.
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8What does the parameter (lambda) represent in a Poisson distribution?
poisson distribution
Easy
A.The probability of success on a single trial.
B.The exact number of events observed.
C.The average number of events in a given interval.
D.The total number of trials.
Correct Answer: The average number of events in a given interval.
Explanation:
The parameter is the rate parameter, which represents the mean or expected number of occurrences of an event in the specified interval of time, space, or volume.
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9Which of the following is NOT a necessary property of a Bernoulli process?
bernoulli process
Easy
A.The probability of success is constant for each trial.
B.Each trial has only two possible outcomes.
C.The trials are independent of each other.
D.The number of successes must equal the number of failures.
Correct Answer: The number of successes must equal the number of failures.
Explanation:
A Bernoulli process requires only three properties: two outcomes per trial, a constant probability of success (), and independence between trials. The number of successes and failures can vary.
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10A key property of the Geometric distribution is that it is 'memoryless'. This implies that:
geometric distribution
Easy
A.The outcome of one trial is determined by the previous trial.
B.The probability of success must be 0.5.
C.The probability of a future success does not depend on past failures.
D.The distribution has no mean or variance.
Correct Answer: The probability of a future success does not depend on past failures.
Explanation:
The memoryless property means that if you haven't succeeded yet, the probability of succeeding on the next attempt is the same as it was on the very first attempt. The past is 'forgotten'.
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11If a random variable follows a Binomial distribution with parameters and , what is the mean (expected value) of ?
binomial distribution
Easy
A.8
B.4.8
C.12
D.20
Correct Answer: 8
Explanation:
The mean of a Binomial distribution is given by the formula . In this case, .
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12The function is the Moment Generating Function (MGF) for which distribution?
moment generating function ( without proof) of the above mentioned distributions
Easy
A.Poisson distribution
B.Bernoulli distribution
C.Geometric distribution
D.Binomial distribution
Correct Answer: Poisson distribution
Explanation:
The standard form of the MGF for a Poisson distribution with rate parameter is .
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13A Negative Binomial distribution describes the probability of the number of...
negative binomial distribution
Easy
A.events occurring in a fixed interval.
B.trials required to obtain a fixed number of successes ().
C.failures before the first success.
D.successes in a fixed number of trials ().
Correct Answer: trials required to obtain a fixed number of successes ().
Explanation:
The Negative Binomial distribution models the waiting time (in terms of trials) to achieve a predetermined number of successes, denoted by .
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14A unique property of the Poisson distribution is that its mean is equal to its...
poisson distribution
Easy
A.mode.
B.standard deviation.
C.variance.
D.median.
Correct Answer: variance.
Explanation:
For a Poisson distribution with parameter , both the mean (expected value) and the variance are equal to . That is, .
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15The expression represents the Moment Generating Function (MGF) of which distribution?
moment generating function ( without proof) of the above mentioned distributions
Easy
A.Geometric distribution
B.Negative Binomial distribution
C.Binomial distribution
D.Poisson distribution
Correct Answer: Binomial distribution
Explanation:
A Binomial random variable is the sum of independent Bernoulli trials. Since the MGF of a sum of independent variables is the product of their MGFs, the Binomial MGF is the Bernoulli MGF, , raised to the power of .
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16A random variable can take a value of 1 with probability and a value of 0 with probability . This random variable follows a...
bernoulli process
Easy
A.Poisson distribution.
B.Binomial distribution with n=1.
C.Bernoulli distribution.
D.Uniform distribution.
Correct Answer: Bernoulli distribution.
Explanation:
This is the definition of a Bernoulli random variable. It represents the outcome of a single trial with two possibilities, typically coded as 1 for success and 0 for failure. Note that a Binomial distribution with n=1 is a Bernoulli distribution.
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17A student takes a 10-question multiple-choice quiz. Each question has 4 options, and the student guesses randomly on every question. Which distribution would model the number of questions the student answers correctly?
binomial distribution
Easy
A.Binomial distribution
B.Negative Binomial distribution
C.Geometric distribution
D.Poisson distribution
Correct Answer: Binomial distribution
Explanation:
This is a Binomial experiment because there is a fixed number of trials (n=10), each guess is independent, there are two outcomes (correct/incorrect), and the probability of success (guessing correctly, p=0.25) is constant.
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18An operator dials a phone number repeatedly until they get a connection. What is the single parameter needed to define the Geometric distribution for this scenario?
geometric distribution
Easy
A.The probability of getting a connection on any trial ().
B.The average number of dials needed.
C.The number of failed attempts.
D.The total number of dials they will make ().
Correct Answer: The probability of getting a connection on any trial ().
Explanation:
The Geometric distribution is completely defined by a single parameter, , which is the constant probability of success on each independent trial.
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19A city emergency service receives an average of 3 calls per hour for a certain type of incident. What is the value of if we want to model the number of calls in a 1-hour interval?
poisson distribution
Easy
A.3
B.1/3
C.1
D.60
Correct Answer: 3
Explanation:
The parameter is the average rate of events for the interval of interest. Since the average is given as 3 calls per hour and the interval is 1 hour, .
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20A quality control inspector checks items from a production line until she finds 5 defective items. Which distribution would model the total number of items she inspects?
negative binomial distribution
Easy
A.Geometric distribution
B.Binomial distribution
C.Negative Binomial distribution
D.Poisson distribution
Correct Answer: Negative Binomial distribution
Explanation:
This scenario involves counting the number of trials (inspected items) needed to reach a fixed number of successes (finding 5 defective items). This is the definition of a Negative Binomial distribution.
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21A random variable follows a binomial distribution, , with a mean of 6 and a variance of 2.4. What are the values of and ?
binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The mean of a binomial distribution is and the variance is . We are given and . Substituting the first equation into the second gives , which solves to , so . Then, using , we find , which gives .
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22A customer support center receives an average of 5 calls per minute. What is the probability of receiving exactly 12 calls in a 2-minute interval, assuming the calls follow a Poisson process?
poisson distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The average rate is 5 calls/minute. For a 2-minute interval, the new average rate is calls. The Poisson probability mass function is . Plugging in and , we get .
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23A basketball player makes 80% of her free throws. What is the probability that her first miss occurs on her 5th attempt?
geometric distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
This is a geometric distribution problem where 'success' is defined as a miss. The probability of a miss is . The probability of the first success (miss) occurring on the trial is given by . Here, this translates to 4 successful free throws (failures in this context) followed by one miss. So, .
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24A quality control inspector is testing light bulbs. The probability that a bulb is defective is 0.1. What is the probability that the 3rd defective bulb found is the 10th bulb tested?
negative binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
This follows a negative binomial distribution with successes (finding a defective bulb) and . We want the probability that the success occurs on the trial. The PMF is . Plugging in the values, we get .
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25The moment generating function (MGF) of a discrete random variable is . What is the variance of ?
moment generating function ( without proof) of the above mentioned distributions
Medium
A.1.8
B.14.0
C.4.2
D.6.0
Correct Answer: 4.2
Explanation:
The MGF is in the form of a binomial distribution, , where . By comparing, we can identify , , and . The variance of a binomial distribution is . So, .
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26If a random variable follows a Poisson distribution such that , what is the variance of ?
poisson distribution
Medium
A.
B.2
C.8
D.4
Correct Answer: 4
Explanation:
For a Poisson distribution, . We are given , so . Simplifying, we get , which leads to . Since , we can divide by to get . For a Poisson distribution, the variance is equal to the mean, so .
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27A student takes a 10-question multiple-choice quiz, with 5 options per question. If the student guesses randomly on all questions, what is the probability of answering at least one question correctly?
binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The number of correct answers follows a binomial distribution . The probability of answering at least one correctly is . It's easier to calculate the complement: . The probability of getting zero questions correct is . Therefore, .
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28Which of the following scenarios best describes a Bernoulli process?
bernoulli process
Medium
A.Flipping a fair coin until you get 3 heads and counting the number of flips.
B.Rolling a fair six-sided die 10 times and recording the number that appears on each roll.
C.Drawing 5 cards from a standard deck without replacement and counting the number of Aces.
D.Asking 20 students their favorite color from a list of 5 colors.
Correct Answer: Flipping a fair coin until you get 3 heads and counting the number of flips.
Explanation:
A Bernoulli process consists of a sequence of independent trials, each with two outcomes (success/failure) and a constant probability of success. Flipping a coin has two outcomes (heads/tails), the probability is constant (0.5), and flips are independent. Drawing cards without replacement violates independence. Rolling a die has more than two outcomes. Asking for a favorite color also has more than two outcomes.
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29The MGF of a random variable Y is . What is ?
moment generating function ( without proof) of the above mentioned distributions
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The MGF is in the form of a Poisson distribution, , with mean . We need to find . For a Poisson distribution, and . Thus, .
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30If is a geometric random variable representing the trial number of the first success and its mean is , what is ?
geometric distribution
Medium
A.0.80
B.0.64
C.0.16
D.0.36
Correct Answer: 0.64
Explanation:
For a geometric distribution, the mean is . Given , we have , so the probability of success is . The probability of failure is . The probability that the first success occurs after the trial is . Therefore, .
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31The number of trials to get the -th success follows a negative binomial distribution. If and , what are the parameters and ?
negative binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
For the negative binomial distribution, and . We have and . We can write the variance as . Substituting , we get , which simplifies to . This gives , so and . Whoops, let's re-calculate. . Then , which isn't an integer. Let's check my formulas. Ah, some definitions use number of failures. Let's use the definition of as number of trials. and is correct. Let me re-check the calculation. . Then . This suggests an error in the question or options. Let's try plugging in the correct option . Then and . This doesn't match the variance of 30. Let's check option A: . . . Let me adjust the question to have a valid answer. Let's set and . Then . Then . Okay, let's create a valid question. Let and . Then . Then . Let me use this. The question will be: If and , find and . Options: A) . B) . C) . D) . The correct option is A.
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32The number of trials to get the -th success follows a negative binomial distribution. If the mean is and the variance is , what are the parameters and ?
negative binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
For the negative binomial distribution (where X is the number of trials), the mean is and variance is . We are given and . We can rewrite the variance formula as . Substituting the given values, we get . Dividing by 20 gives , which solves to , so and . Now use the mean formula: , which gives .
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33The number of flaws in a fiber optic cable follows a Poisson distribution with a mean of 0.6 per 100 feet. What is the probability that there are no flaws in a 500-foot cable?
poisson distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The mean rate is 0.6 flaws per 100 feet. For a 500-foot cable, the new mean is . The probability of having flaws is given by the Poisson PMF . With , this becomes .
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34In a large population, 40% of people have blood type A. If 5 people are chosen at random, what is the probability that exactly 2 of them have blood type A?
binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
This scenario follows a binomial distribution with trials (people) and probability of success (having blood type A). We want to find the probability of exactly successes. The binomial PMF is . Plugging in the values gives .
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35The moment generating function of a random variable X is . What kind of distribution does X follow and what is its mean?
moment generating function ( without proof) of the above mentioned distributions
Medium
A.Poisson, Mean = 4
B.Negative Binomial, Mean = 6.67
C.Binomial, Mean = 2.4
D.Geometric, Mean = 1.67
Correct Answer: Negative Binomial, Mean = 6.67
Explanation:
The MGF has the form , which corresponds to a negative binomial distribution. By comparing, we can identify , , and . The mean of a negative binomial distribution (representing the number of trials) is . Therefore, the mean is .
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36A machine produces items with a 5% defect rate. Items are inspected one by one. What is the probability that the first defective item is found within the first 3 items inspected?
geometric distribution
Medium
A.0.05
B.0.142625
C.0.15
D.1 - (0.05)^3
Correct Answer: 0.142625
Explanation:
This is a geometric distribution with . We want to find . The PMF is . \ . \ . \ . \ The sum is . Alternatively, .
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37Which of these statements correctly describes a key difference between the Binomial and Poisson distributions?
poisson distribution
Medium
A.The Binomial distribution is for discrete variables, while the Poisson is for continuous variables.
B.The probability of success must be constant for a Poisson distribution, but can vary in a Binomial distribution.
C.The Binomial distribution has a fixed number of trials, while the Poisson distribution describes events over a continuous interval.
D.The mean and variance are always equal in a Binomial distribution, but not in a Poisson distribution.
Correct Answer: The Binomial distribution has a fixed number of trials, while the Poisson distribution describes events over a continuous interval.
Explanation:
The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials (). The Poisson distribution models the number of occurrences of an event in a fixed interval of time, space, or volume, where there is no theoretical upper limit to the number of events.
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38A coin is biased so that the probability of heads is . What is the variance of the number of tosses required to get the 4th head?
negative binomial distribution
Medium
A.9.33
B.44.44
C.13.33
D.2.8
Correct Answer: 44.44
Explanation:
This describes a negative binomial distribution where is the number of tosses (trials) to get successes (heads). The probability of success is . The variance is given by the formula . Plugging in the values, we get .
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39It is known that 10% of a certain brand of computer chips are defective. If a company orders 15 chips, what is the probability that at most one chip is defective?
binomial distribution
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The number of defective chips follows a binomial distribution with and . 'At most one' means . \ . \ . \ The total probability is the sum of these two values.
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40A random variable X has a moment generating function . Which statement is correct?
moment generating function ( without proof) of the above mentioned distributions
Medium
A.X is a Geometric random variable with a mean of 0.2.
B.X is a Binomial random variable with n=1.
C.X is a Geometric random variable with a mean of 5.
D.X is a Negative Binomial random variable with r=1 and p=0.8.
Correct Answer: X is a Geometric random variable with a mean of 5.
Explanation:
The MGF is in the form of a geometric distribution, , where . By comparing, we see and . A random variable with this MGF follows a geometric distribution. The mean of a geometric distribution is . Therefore, the mean is .