In a binomial experiment, the probability of success stays the same from trial to trial.
Correct Answer
True
True. A binomial experiment requires identical trials, so p is the same on every trial and q = 1 - p is the probability of failure on every trial. If p changed between trials, the binomial formulas would not apply.
Category: Binomial Distribution
Question 2
In a binomial experiment, the number of trials is set by when the first success occurs.
Correct Answer
False
False. In a binomial experiment the number of trials, n, is fixed in advance. Counting trials until the first success describes a geometric experiment instead.
Category: Binomial Distribution
Question 3
The variance of a binomial random variable equals npq, where q = 1 - p.
Correct Answer
True
True. A binomial random variable has variance npq and standard deviation sqrt(npq). For example, with n = 10 and p = 0.5 the variance is 10(0.5)(0.5) = 2.5.
Category: Binomial Distribution
Question 4
The mean of a binomial distribution with n = 10 and p = 0.4 is 6.
Correct Answer
False
False. The binomial mean is np = 10(0.4) = 4. The value 6 is nq = 10(0.6), the expected number of failures.
Category: Binomial Distribution
Question 5
If X has a binomial distribution with n trials, X counts the number of successes in those n trials.
Correct Answer
True
True. A binomial random variable is the count of successes in a fixed number of independent trials. Its possible values are the whole numbers from 0 to n.
Category: Binomial Distribution
Question 6
A geometric random variable that counts trials up to and including the first success can take the value 0.
Correct Answer
False
False. At least one trial is needed to get the first success, so this geometric random variable takes the values 1, 2, 3 and so on. The value 0 is possible only for the version that counts failures before the first success.
Category: Geometric Distribution
Question 7
In a hypergeometric experiment, X counts how many items from the group of interest appear in the sample.
Correct Answer
True
True. A hypergeometric experiment samples without replacement from two groups, and X is the number of sampled items that come from the group of interest. Its mean is the sample size times the share of the population in that group.
Category: Hypergeometric Distribution
Question 8
If calls arrive at an average rate of 6 per hour, the Poisson mean for a 30-minute period is 6.
Correct Answer
False
False. The Poisson mean must match the length of the interval. Half an hour at 6 calls per hour gives a mean of 3 calls for the 30-minute period.
Category: Poisson Distribution
Question 9
For a Poisson distribution, the mean and the variance are equal.
Correct Answer
True
True. A Poisson random variable with mean mu also has variance mu, so its standard deviation is sqrt(mu). For example, a mean of 16 gives a standard deviation of 4.
Category: Poisson Distribution
Question 10
A Poisson random variable can take only values from 0 up to its mean.
Correct Answer
False
False. A Poisson random variable can take any whole number 0, 1, 2, 3 and beyond, with no upper limit. Values above the mean are common; the mean is only the average count per interval.
Category: Poisson Distribution
Question 11
The variance of a discrete random variable is found by adding (x - mu)^2 times P(x) over all values of x.
Correct Answer
True
True. Each squared distance from the mean is weighted by its probability, and the weighted squared distances are added. The standard deviation is the square root of that sum.
Category: Mean and Standard Deviation
Question 12
The expected value of a discrete random variable must be one of the values the variable can take.
Correct Answer
False
False. The expected value is a long-run average and does not have to be a possible value. For example, a variable taking only the values 0, 1 and 2 can have an expected value of 0.9.
Category: Mean and Standard Deviation
Question 13
A larger standard deviation means the values of a random variable tend to fall farther from its mean.
Correct Answer
True
True. The standard deviation measures the typical distance of the values from the mean. A larger standard deviation means more spread, and a smaller one means values cluster near the mean.
Category: Mean and Standard Deviation
Question 14
In a discrete probability distribution, a single probability may be greater than 1 as long as all of them add to 1.
Correct Answer
False
False. Every probability in a discrete distribution must be between 0 and 1, inclusive, and together they must add to 1. If one probability were above 1, some other probability would have to be negative.
Category: Probability Distributions
Question 15
A binomial distribution with n = 10 and p = 0.5 has its highest probability at X = 5.
Correct Answer
True
True. With p = 0.5 the binomial distribution is symmetric about its mean, np = 5, and the peak is at 5. For example, P(X = 5) is about 0.246, larger than the probability of any other value.
Category: Binomial Distribution
Question 16
The number of defective parts in a sample of 3 drawn without replacement from a box of 12 parts follows a binomial distribution exactly.
Correct Answer
False
False. Drawing without replacement from a small population changes the chances after each draw, so the trials are not independent. The count follows a hypergeometric distribution.
Category: Hypergeometric Distribution
Question 17
A geometric experiment repeats independent trials, each with the same probability of success, until the first success occurs.
Correct Answer
True
True. The geometric distribution models how many trials it takes to reach the first success. Each trial is independent and has the same probability of success, p.
Category: Geometric Distribution
Question 18
In a geometric experiment, the probability of success grows with each failed trial.
Correct Answer
False
False. The trials in a geometric experiment are independent with a constant probability of success, p. A run of failures does not make success more likely on the next trial.
Category: Geometric Distribution
Question 19
Over many plays, a game with a negative expected value for the player loses money for the player on average.
Correct Answer
True
True. The expected value is the long-run average gain per play. A negative expected value means the average result over many plays is a loss, even though single plays can win.
Category: Mean and Standard Deviation
Question 20
If X has a binomial distribution with n = 8, X can take any whole number from 1 to 8.
Correct Answer
False
False. A binomial random variable can take any whole number from 0 to n, so with n = 8 the possible values are 0 through 8. Zero successes is always possible when p is less than 1.
Category: Binomial Distribution