Question 1

Which of the following is a discrete variable?

Accepted Answer

The number of cars sold by a dealership in a day

Question 2

A probability distribution for the number of network interruptions per day is given as follows: 0 (0.35), 1 (0.25), 2 (0.20), 3 (0.10), 4 (0.05), 5 (0.05). What is the probability of having at least 3 interruptions in a day?

Accepted Answer

$$0.20$$

Question 3

Which formula correctly represents the expected value (mean) of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$P(X = x_i)$$?

Accepted Answer

$$\mu = E(X) = \sum_{i=1}^N x_i P(X = x_i)$$

Question 4

A company contacts 20 customers in an email campaign. Each customer has a 30% chance of making a purchase. What is the expected number of purchases?

Accepted Answer

$$6$$

Question 5

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

Accepted Answer

The number of customer complaints received per day

Question 6

Given a Poisson process with $$\lambda = 3$$, what is the probability of exactly 2 events occurring in a given interval?

Accepted Answer

$$P(X=2) = \frac{e$$^{-3}$$ 3^2$}{2!}$$

Question 7

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

Accepted Answer

Each probability must be greater than 1

Question 8

A support center receives an average of 4 tickets per hour. What is the standard deviation of the number of tickets per hour, assuming a Poisson distribution?

Accepted Answer

$$2$$

Question 9

Which formula gives the variance of a binomial distribution with parameters $$n$$ and $$p$$?

Accepted Answer

$$\$$sigma^2$$ = n p (1-p)$$

Question 10

A manager wants to know the risk associated with two projects. Both have the same expected profit, but Project A has a higher standard deviation than Project B. What does this imply?

Accepted Answer

Project A is riskier because its outcomes are more dispersed

Question 11

Which formula correctly gives the probability of exactly $$x$$ successes in $$n$$ independent trials with probability $$p of $$success in each trial (binomial probability)?

Accepted Answer

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

Question 12

Which of the following best describes the difference between expected value and the probability of a specific outcome?

Accepted Answer

Expected value is the long-run average; probability of a specific outcome is the chance of that exact result

Question 13

A business wants to model the number of website crashes per week. Which distribution should they use?

Accepted Answer

Poisson

Question 14

If $$X is a $$discrete random variable with $$E(X) = 2$$ and $$Var(X) = 3$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{3}$$

Question 15

Which of the following is NOT a condition for using the binomial distribution?

Accepted Answer

The number of trials is unknown

Question 16

A company wants to know the probability that exactly 1 out of 4 invoices is late, given the probability of a late invoice is 0.10. Which calculation is correct?

Accepted Answer

$$P(X=1) = \frac{4!}{1!3!} (0.10)^1$ (0.90)^3$

Question 17

Which of the following is a real-world example where a binomial distribution is appropriate?

Accepted Answer

Counting the number of defective items in a batch of 30

Question 18

If the mean number of events per interval in a Poisson process is $$\lambda$$, what is the variance?

Accepted Answer

$$\lambda$$

Question 19

Which statement about the standard deviation of a discrete probability distribution is correct?

Accepted Answer

It measures the typical deviation from the expected value

Discrete Probability Distributions: Binomial, Poisson, Expected Value, and Risk

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