Question 1

Which of the following is a key difference between discrete and continuous random variables?

Accepted Answer

Discrete random variables can only take on a finite or countable number of values, while continuous random variables can take on any value within an interval.

Question 2

Which of the following is NOT a requirement for a discrete probability distribution?

Accepted Answer

The sum of all probabilities must be less than 1.

Question 3

Given the following probability distribution for the number of heads in two coin flips: $$P(0) = 0.25$$, $$P(1) = 0.5$$, $$P(2) = 0.25$$, what is the mean (expected value) of the distribution?

Accepted Answer

$$\mu = 1.00$$

Question 4

Which formula correctly calculates the variance of a discrete probability distribution?

Accepted Answer

$$\$$sigma^2$$ = \sum_{i=1}^n (x_i - \mu)^2$ P(x_i)$$

Question 5

A business wants to know the expected profit from a new product. The possible profits and their probabilities are: $$\0$ (0.2), $$\$$1000 (0.5$$), $$\2000$ (0.3). What is the expected monetary value (EMV)?

Accepted Answer

$$EMV = \1100$

Question 6

Which of the following scenarios best fits a binomial experiment?

Accepted Answer

Testing 20 light bulbs to see if each is defective or not.

Question 7

What is the formula for the probability of exactly $$x$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

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

Question 8

If 40% of voters support a proposition and 10 voters are polled, what is the mean and standard deviation of the number of supporters?

Accepted Answer

$$\mu = 4$$, $$\sigma = 1.55$$

Question 9

Which Excel function would you use to find the probability of exactly $$x$$ successes in a binomial distribution?

Accepted Answer

BINOM.DIST(x, n, p, FALSE)

Question 10

A Poisson distribution is most appropriate for which of the following scenarios?

Accepted Answer

Counting the number of emails received per hour.

Question 11

What is the formula for the Poisson probability of observing $$x$$ events when the mean number of events is $$\lambda$$?

Accepted Answer

$$P(X = x) = \frac{e$$^{-\lambda}$$ \lambda^x}{x!}$$

Question 12

If a bank receives an average of 6 customer arrivals every 30 minutes, what is the probability that exactly 4 customers will arrive in the next 30 minutes? (Use the Poisson formula.)

Accepted Answer

$$P(X = 4) = \frac{e$$^{-6}$$ 6^4$}{4!}$$

Question 13

Under what conditions can the Poisson distribution be used to approximate the binomial distribution?

Accepted Answer

When $$n \geq 20$$ and $$p \leq 0.05$$

Question 14

Which of the following best describes a hypergeometric distribution scenario?

Accepted Answer

Selecting a sample of products from a batch without replacement to check for defects.

Question 15

What is the formula for the probability of $$x$$ successes in a hypergeometric distribution?

Accepted Answer

$$P(X = x) = \frac{\binom{R}{x} \binom{N-R}{n-x}}{\binom{N}{n}}$$

Question 16

A population contains 50 accounts, 5 of which are delinquent. If 10 accounts are selected at random without replacement, what is the expected number of delinquent accounts in the sample?

Accepted Answer

1.0

Question 17

Which Excel function is used to calculate hypergeometric probabilities?

Accepted Answer

HYPGEOM.DIST

Question 18

If a discrete probability distribution has a mean $$\mu = 2$$ and variance $$\$$sigma^2$$ = 0.5$$, what is the standard deviation?

Accepted Answer

$$\sigma = 0.707$$

Discrete Probability Distributions: Binomial, Poisson, and Hypergeometric

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