Quiz 2: Discrete Distributions 📊

Test your knowledge of the material on discrete distributions in the following quiz to see how much you learned. This is entirely private for you---no records are kept of your performance.

Questions

1. What are the assumptions necessary for a variable to be distributed as a binomial distribution?

The variable must be binary The variable must be discrete The variable must have a fixed number of trials All of the above

Explanation

A binomial distribution requires ALL of these assumptions: the variable must be binary (success/failure), discrete (counts), and have a fixed number of trials, along with independent trials and constant probability.

2. How does the skewness of a binomial variable relate to its parameters n and p?

Skewness = n p Skewness = n/p Skewness = p (1-p) / n Skewness = p^2 / n

Explanation

The skewness of a binomial distribution is approximately p(1-p)/n. This shows that skewness decreases as n increases, and is minimized when p=0.5 (symmetric distribution).

3. When would you use a negative binomial distribution over a Poisson distribution?

When the number of trials is fixed When the variance exceeds the mean When there are negative counts in the data When the parameter p differs substantially from 0.5

Explanation

The key difference is overdispersion: use negative binomial when variance > mean. The Poisson distribution assumes mean = variance, which is often violated in real count data.

4. How does the mean and variance of a Poisson distribution relate to its parameter λ?

Mean = λ, Variance = λ Mean = 1/λ, Variance = 1/λ^2 Mean = 1/λ, Variance = λ Mean = λ, Variance = λ^2

Explanation

For the Poisson distribution, both the mean and variance equal the parameter λ. This equidispersion property (mean = variance) is a key characteristic of the Poisson distribution.

5. Which example is most likely to follow a log-series distribution?

Number of accidents at a busy intersection Counts of occurrences of distinct words in the books written by Jane Austen Number of births of males in Canadian families composed of 6 children Number of tosses of a coin required to get the first head

Explanation

The log-series distribution is best for highly skewed count data where there are many low-frequency events and few high-frequency events, such as word frequencies in text (following Zipf's law). Option (a) would be Poisson, (c) would be binomial, and (d) would be geometric.

6. Which of the following characterize the negative binomial distribution and the geometric distribution?

The negative binomial distribution models the number of failures before a specified number of successes The geometric distribution models the number of failures before the first success Both A and B Neither A nor B

Explanation

Both statements are correct. The negative binomial distribution counts failures before achieving r successes, and the geometric distribution is the special case where r=1, counting failures before the first success.

7. In what type of data might you use a negative binomial distribution?

Count data Binary data Continuous data Both A and B

Explanation

The negative binomial distribution is used for count data (non-negative integers), particularly when the data show overdispersion. It is not appropriate for binary data (that would be binomial or Bernoulli) or continuous data.

8. What is the mean and variance of a variable distributed as a geometric distribution?

Mean = λ, Variance = λ Mean = 1/λ, Variance = 1/λ^2 Mean = 1/λ, Variance = λ Mean = λ, Variance = 1/λ^2

Explanation

For the geometric distribution with parameter p, the mean is (1-p)/p and variance is (1-p)/p^2. Note: There are two parameterizations of the geometric; DDAR uses parameter p rather than λ. This question is problematic as noted in the original quiz.