1Which situation can be modeled using a binomial distribution?
Binomial Distribution and its properties
Easy
A.Counting successes in a fixed number of independent trials
B.Counting trials required to obtain a fixed number of successes
C.Measuring the height of randomly selected students
D.Measuring the waiting time between two events
Correct Answer: Counting successes in a fixed number of independent trials
Explanation:
A binomial distribution counts successes in a fixed number of independent trials with constant success probability.
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2If , what is the mean of ?
Binomial Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The mean of a binomial random variable is .
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3If , what is the variance of ?
Binomial Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The variance of a binomial random variable is .
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4In a binomial experiment, how many possible outcomes does each trial have?
Binomial Distribution and its properties
Easy
A.Four
B.Three
C.One
D.Two
Correct Answer: Two
Explanation:
Each binomial trial has two possible outcomes, commonly called success and failure.
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5Which expression gives for ?
Binomial Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The binomial probability mass function is .
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6A Poisson distribution is commonly used to model what quantity?
Poisson Distribution and its properties
Easy
A.The value of a continuous measurement
B.The number of events in a fixed interval
C.The number of successes in fixed trials
D.The proportion of successful outcomes
Correct Answer: The number of events in a fixed interval
Explanation:
A Poisson distribution models the number of events occurring in a fixed interval of time, area, or space.
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7If , what is the mean of ?
Poisson Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The mean of a Poisson random variable is .
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8If , what is the variance of ?
Poisson Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
For a Poisson distribution, both the mean and variance are equal to .
Incorrect! Try again.
9Which expression is the probability mass function of a Poisson random variable?
Poisson Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The Poisson probability mass function is .
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10What values can a Poisson random variable take?
Poisson Distribution and its properties
Easy
A.All real numbers
B.All integers
C.All positive real numbers
D.All nonnegative integers
Correct Answer: All nonnegative integers
Explanation:
A Poisson variable is a count, so it can take the values .
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11What does a negative binomial distribution commonly model?
Negative Binomial Distribution and its properties
Easy
A.Events occurring in a fixed interval
B.Successes in a fixed number of trials
C.Measurements clustered around a mean
D.Failures before a fixed number of successes
Correct Answer: Failures before a fixed number of successes
Explanation:
A negative binomial distribution commonly counts the failures observed before a specified number of successes occurs.
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12Which special case of the negative binomial distribution occurs when the required number of successes is ?
Negative Binomial Distribution and its properties
Easy
A.Geometric distribution
B.Normal distribution
C.Poisson distribution
D.Binomial distribution
Correct Answer: Geometric distribution
Explanation:
When only one success is required, the negative binomial distribution reduces to the geometric distribution.
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13In a negative binomial experiment, what must remain constant from trial to trial?
Negative Binomial Distribution and its properties
Easy
A.The success probability
B.The number of failures
C.The trial duration
D.The observed outcome
Correct Answer: The success probability
Explanation:
Negative binomial trials are independent Bernoulli trials with the same success probability .
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14Let count failures before successes, with success probability . What is the mean of ?
Negative Binomial Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
When counts failures before successes, its mean is .
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15Let count failures before successes, with success probability . What is the variance of ?
Negative Binomial Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
For this form of the negative binomial distribution, .
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16What is the general shape of a normal distribution?
Normal Distribution and its properties
Easy
A.Triangular and skewed
B.U-shaped and symmetric
C.Bell-shaped and symmetric
D.Flat and symmetric
Correct Answer: Bell-shaped and symmetric
Explanation:
A normal distribution has a symmetric, bell-shaped probability density curve.
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17For a normal distribution, how are the mean, median, and mode related?
Normal Distribution and its properties
Easy
A.Median is greater than both
B.Mode is greater than both
C.Mean is greater than both
D.All three are equal
Correct Answer: All three are equal
Explanation:
Because a normal distribution is symmetric and unimodal, its mean, median, and mode are equal.
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18What are the mean and standard deviation of the standard normal distribution?
Normal Distribution and its properties
Easy
A.Mean and standard deviation
B.Mean and standard deviation
C.Mean and standard deviation
D.Mean and standard deviation
Correct Answer: Mean and standard deviation
Explanation:
The standard normal distribution has mean and standard deviation .
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19Approximately what percentage of observations in a normal distribution lie within one standard deviation of the mean?
Normal Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The empirical rule states that about of observations lie within one standard deviation of the mean.
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20What is the total area under a normal distribution curve?
Normal Distribution and its properties
Easy
A.
B.
C.
D.
Correct Answer:
Explanation:
The total area under any probability density curve, including the normal curve, is .
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21A salesperson independently closes each call with probability . If 10 calls are made, what is the probability of closing exactly 3 calls?
Binomial Distribution and its properties
Medium
A.
B., which represents the probability of closing no more than three calls
C.
D.
Correct Answer:
Explanation:
For , .
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22A binomial random variable has mean and variance . What are its parameters and ?
Binomial Distribution and its properties
Medium
A.
B.
C.
D., since the variance is treated as the binomial mean
Correct Answer:
Explanation:
Since and , dividing gives . Thus and .
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23Each manufactured component is defective with probability , independently of the others. What is the probability that a batch of 20 components contains at least one defective component?
Binomial Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Use the complement: .
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24If , what is the ratio ?
Binomial Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The ratio is .
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25Let and be independent. What is the distribution of ?
Binomial Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
Independent binomial variables with the same success probability add by combining their trial counts, so .
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26A help desk receives an average of 4 calls every 10 minutes. Assuming a Poisson model, what is the probability of receiving exactly 2 calls in a 10-minute interval?
Poisson Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
For , .
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27Customers arrive at a store according to a Poisson process at an average rate of 6 per hour. What is the probability that no customer arrives during a 30-minute period?
Poisson Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The mean for 30 minutes is . Therefore, .
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28A Poisson random variable has variance . What are its mean and standard deviation?
Poisson Distribution and its properties
Medium
A.Mean , standard deviation
B.Mean , standard deviation
C.Mean , standard deviation
D.Mean , standard deviation , because variance is the square of the parameter
Correct Answer: Mean , standard deviation
Explanation:
For a Poisson distribution, both the mean and variance equal . Thus the mean is and the standard deviation is .
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29Independent random variables and have Poisson distributions with means and , respectively. What is ?
Poisson Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The sum is Poisson with mean . Hence .
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30For a Poisson random variable , it is known that . What is the value of the Poisson parameter ?
Poisson Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
For a Poisson variable, . Equality of the probabilities gives .
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31Let be the number of failures before the third success in independent trials with success probability . What is ?
Negative Binomial Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
For failures before the third success, .
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32Let count the number of failures before the fourth success when the success probability is . What are the mean and variance of ?
Negative Binomial Distribution and its properties
Medium
A.Mean , variance
B.Mean , variance
C.Mean , variance
D.Mean , variance
Correct Answer: Mean , variance
Explanation:
For failures before successes, and .
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33Independent trials have success probability . What is the expected total number of trials needed to obtain 5 successes?
Negative Binomial Distribution and its properties
Medium
A.
B., obtained by multiplying the required successes by the success probability
C.
D.
Correct Answer:
Explanation:
The expected number of trials required for successes is . Thus the expectation is .
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34A basketball player makes each free throw independently with probability . What is the probability that the player's third successful throw occurs on the fifth attempt?
Negative Binomial Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
There must be exactly 2 successes in the first 4 attempts and a success on the fifth: .
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35Suppose and are independent numbers of failures before 2 and 3 successes, respectively, with the same success probability . What is the distribution of ?
Negative Binomial Distribution and its properties
Medium
A.Trials before 5 failures with success probability
B.Failures before 5 successes with probability
C.Failures before 6 successes with probability
D.Failures before 5 successes with probability
Correct Answer: Failures before 5 successes with probability
Explanation:
Independent negative binomial variables with the same combine by adding their required success counts. Therefore, counts failures before successes.
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36Test scores are normally distributed with mean and standard deviation . Approximately what proportion of students score above ?
Normal Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The standardized value is . Thus .
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37The weights of packages are normally distributed with mean g and standard deviation g. Approximately what weight marks the 90th percentile? Use .
Normal Distribution and its properties
Medium
A. g
B. g
C. g
D. g
Correct Answer: g
Explanation:
The 90th-percentile weight is g.
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38A measurement is normally distributed with mean and standard deviation . Which interval contains approximately the central of measurements?
Normal Distribution and its properties
Medium
A.
B.
C.
D., using four standard deviations on each side of the mean
Correct Answer:
Explanation:
The central interval is approximately , giving .
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39If , where the second parameter is the variance, and , what is the distribution of ?
Normal Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
A linear transformation remains normal. Its mean is , and its variance is .
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40The lifetime of a device is normally distributed with mean months and standard deviation months. Approximately what is ?
Normal Distribution and its properties
Medium
A.
B.
C.
D.
Correct Answer:
Explanation:
The bounds correspond to and . Therefore, .
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41Let with . If , what is ?
Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Equating the two probabilities gives , so . Therefore, .
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42Let . Conditional on , each of the successes is independently retained with probability , and is the number retained. For , what is the conditional distribution of given ?
Binomial Distribution and its properties
Hard
A.
B.
C., since conditioning is assumed to leave the original success probability unchanged
D.
Correct Answer:
Explanation:
Each trial is retained with probability and is an unretained success with probability . Given that a trial is not retained, the latter probability becomes .
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43Suppose and are independent, where . If has a binomial distribution, which statement must hold?
Binomial Distribution and its properties
Hard
A., and
B.No restriction on and is required
C., and the sum is binomial with success probability equal to the weighted average of and
D., and
Correct Answer: , and
Explanation:
The probability-generating function is . For it to be a single binomial generating function, the two linear factors must be identical, so .
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44A random variable has exactly two modes, at and . What is the ratio ?
Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Two modes at and imply , so . Hence .
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45For , where may be chosen anywhere in , what is the maximum possible value of ?
Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
For fixed , is maximized at . Here the maximizing value is .
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46Let and be independent. What is ?
Poisson Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Conditionally, . Thus the probability is .
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47Let . Each of the events independently receives label with probability and label with probability , with the two labeling decisions independent for each event. If and count labels and , respectively, what is ?
Poisson Distribution and its properties
Hard
A., because only events receiving neither label contribute to the dependence
B.
C.
D.
Correct Answer:
Explanation:
Events receiving both labels form a shared Poisson component with mean . This common component produces covariance .
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48Suppose and . What is ?
Poisson Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The probability ratio gives , so . Since , the conditional probability is .
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49A Poisson random variable has exactly two modes, at and . What is ?
Poisson Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Two modes at and imply the integer parameter . Therefore, .
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50Let . If , what is ?
Poisson Distribution and its properties
Hard
A.
B.
C.
D., obtained by treating the odd-parity probability as though it were the single probability
Correct Answer:
Explanation:
For a Poisson variable, . Thus , giving .
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51Let be the number of failures before the th success, with independent success probability . If and , what is ?
Negative Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Using and gives and . Zero failures requires consecutive successes.
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52Let and be independent numbers of failures before the nd and rd successes, respectively, with the same success probability . What is ?
Negative Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Conditioning on the sum cancels all powers of and . The probability is .
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53Let count failures before the th success, where is a positive integer and . For which value of does the identity hold for all nonnegative integers ?
Negative Binomial Distribution and its properties
Hard
A.Every , because waiting for several successes preserves the same lack-of-memory property after any number of failures
B. only
C.Every positive integer
D. only
Correct Answer: only
Explanation:
When , is geometric and has the memoryless property. Negative binomial distributions with are not memoryless.
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54Let count failures before the th success when the success probability is . Which set gives all modes of ?
Negative Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
For this parameterization, two adjacent modes occur when is an integer. Here it equals , so the modes are and .
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55Suppose has a gamma distribution with shape and rate , and . What is ?
Negative Binomial Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The Poisson-gamma mixture is negative binomial with success probability . Hence .
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56Let and be independent standard normal variables. What is ?
Normal Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The variables and are jointly normal with zero covariance, so they are independent. Conditioning on does not change .
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57Let be bivariate normal with , , , , and correlation . Using notation, what is the distribution of ?
Normal Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The conditional mean is , and the conditional variance is .
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58Suppose and are independent, where the second parameter is the variance. What is ?
Normal Distribution and its properties
Hard
A.
B., after adding an extra covariance term despite independence
C.
D.
Correct Answer:
Explanation:
The difference is normal with mean and variance . Thus .
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59If , what is , expressed using the standard normal density and distribution function ?
Normal Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
Integration by parts gives . Dividing by and setting gives the result.
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60A normal variable satisfies and , where is the standard normal distribution function. What is ?
Normal Distribution and its properties
Hard
A.
B.
C.
D.
Correct Answer:
Explanation:
The two quantiles imply and . Standardizing and gives and , respectively.
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