Question 1

Which of the following is a requirement for a probability distribution?

Accepted Answer

Both $$0 \leq P(x) \leq 1$$ for each $$x$$ and $$\sum P(x) = 1$$

Question 2

A random variable $$X is $$defined as the number of heads in 3 coin tosses. Which type of random variable is $$X$$?

Accepted Answer

Discrete random variable

Question 3

Which of the following scenarios is best modeled by a Poisson distribution?

Accepted Answer

The number of emails received per hour at a help desk

Question 4

Given the probability distribution below, what is the mean $$\mu$$?

$$\begin{align*}
X & : 0 \quad 1 \quad 2 \
P(X) & : 0.2 \quad 0.5 \quad 0.3
\end{align*}$$

Accepted Answer

$$\mu = 1.1$$

Question 5

Which of the following is NOT a criterion for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial

Question 6

Suppose $$X is a $$binomial random variable with $$n=10$$ and $$p=0.3. $$What is the variance $$\$$sigma^2$$ of X$?

Accepted Answer

$$\$$sigma^2$$ = 2.1$$

Question 7

A probability histogram is most similar to which of the following?

Accepted Answer

A relative frequency histogram

Question 8

Which formula correctly gives the probability of exactly $$k$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

$$P(X = k) = \frac{n!}{k!(n-k)!} p^k (1-p$$)^{n-k}$$

Question 9

A Poisson distribution is appropriate when:

Accepted Answer

Events occur randomly and independently over a fixed interval

Question 10

If $$X is a $$Poisson random variable with mean $$\lambda = 4$$, what is $$P(X = 2)$$?

Accepted Answer

$$P(X=2) = \frac{4^2$ e$$^{-4}$$}{2!}$$

Question 11

Which of the following is a 'significantly high' number of successes according to the rule of thumb discussed in the materials?

Accepted Answer

A value that is more than two standard deviations above the mean

Question 12

A probability distribution is valid if:

Accepted Answer

All probabilities are positive and sum to 1

Question 13

Which of the following is an example of a continuous random variable?

Accepted Answer

The height of a randomly selected student

Question 14

Suppose a probability distribution is given by $$P(x) = \frac{x}{15}$$ for $$x = 0, 1, 2, 3, 4, 5. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and sum to 1

Question 15

In a binomial experiment, which of the following must remain constant for all trials?

Accepted Answer

The probability of success

Question 16

A hospital records the number of babies born each day. Which distribution is most appropriate to model the probability of exactly 3 babies being born tomorrow?

Accepted Answer

Poisson distribution

Question 17

If $$X is a $$binomial random variable with $$n=20$$ and $$p=0.4$$, what is the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{20 	imes 0.4 	imes 0.6}$$

Question 18

Which of the following is NOT a requirement for a Poisson distribution?

Accepted Answer

The number of trials is fixed

Question 19

A survey asks 100 people how many cups of coffee they drink per day. The mean is found to be 2.5 cups with a standard deviation of 0.8 cups. According to the rule of thumb, what is the minimum number of cups that would be considered 'significantly high'?

Accepted Answer

$$2.5 + 2 	imes 0.8 = 4.1$$ cups

Question 20

Which of the following best describes the difference between a discrete and a continuous random variable?

Accepted Answer

A discrete random variable can take on only specific, countable values, while a continuous random variable can take on any value within an interval.

Discrete Probability Distributions: Concepts, Parameters, and Applications

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