Q: A company offers three types of insurance policies: Basic ($100), Standard ($150), and Premium ($200). The probabilities of selling each are 0.2, 0.5, and 0.3, respectively. What is the expected revenue from selling one policy?

$E(X) = 100 \times 0.2 + 150 \times 0.5 + 200 \times 0.3 = 155$

Q: A business wants to know the probability that at least 2 out of 6 randomly selected customers will make a purchase, given the probability of purchase is 0.3. Which rule can be used to simplify this calculation?

The complement rule: $P(\text{at least 2}) = 1 - P(0) - P(1)$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values are based on the outcomes of a random event and can only take on specific, countable values.

Question 2

Suppose you are analyzing the number of defective products in a batch of 10, where each product has a 5% chance of being defective. Which probability distribution is most appropriate for modeling this scenario?

Accepted Answer

Binomial distribution

Question 3

Given a discrete random variable $X$ with possible values $x_i$ and probabilities $p(x_i)$, which formula correctly calculates the expected value $E(X)$?

Accepted Answer

$E(X) = \sum x_i p(x_i)$

Question 4

A company offers three types of insurance policies: Basic ($100), Standard ($150), and Premium ($200). The probabilities of selling each are 0.2, 0.5, and 0.3, respectively. What is the expected revenue from selling one policy?

Accepted Answer

$E(X) = 100 	imes 0.2 + 150 	imes 0.5 + 200 	imes 0.3 = 155$

Question 5

If a random variable $X$ is multiplied by a constant $a$, which formula gives the variance of the new variable $aX$?

Accepted Answer

$Var(aX) = a^2 Var(X)$

Question 6

A Bernoulli trial is characterized by which of the following?

Accepted Answer

Only two possible outcomes: success and failure

Question 7

For a binomial distribution with $n$ trials and probability of success $p$, what is the formula for the probability of exactly $k$ successes?

Accepted Answer

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

Question 8

A student scores 88 on a test where the mean is 84 and the standard deviation is 8. What is the student's z-score?

Accepted Answer

$z = \frac{88 - 84}{8} = 0.5$

Question 9

Which of the following is true about the standard normal distribution?

Accepted Answer

It has a mean of 0 and a standard deviation of 1.

Question 10

If $X$ and $Y$ are independent random variables, what is the variance of their sum?

Accepted Answer

$Var(X + Y) = Var(X) + Var(Y)$

Question 11

A continuous uniform distribution is defined on the interval $[a, b]$. What is the formula for its probability density function (PDF)?

Accepted Answer

$f(x) = \frac{1}{b-a}$ \, for \, a \leq x \leq b$

Question 12

In a geometric distribution, what does the expected value $E(X)$ represent?

Accepted Answer

The expected number of trials until the first success.

Question 13

A business wants to know the probability that at least 2 out of 6 randomly selected customers will make a purchase, given the probability of purchase is 0.3. Which rule can be used to simplify this calculation?

Accepted Answer

The complement rule: $P(	ext{at least 2}) = 1 - P(0) - P(1)$

Question 14

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is discrete.

Question 15

If a constant $c$ is added to every value of a random variable $X$, what happens to the variance?

Accepted Answer

The variance remains unchanged.

Question 16

A uniform discrete distribution is defined for $X = 1, 2, 3, 4, 5, 6$. What is the probability of $X = 4$?

Accepted Answer

$P(X = 4) = \frac{1}{6}$

Question 17

Which formula correctly calculates the standard deviation of a discrete random variable $X$?

Accepted Answer

$\sigma_X = \sqrt{Var(X)}$

Question 18

A business analyst wants to model the time until the first customer arrives at a store. Which probability distribution is most appropriate?

Accepted Answer

Exponential distribution

Question 19

If $X$ and $Y$ are correlated random variables, which formula gives the variance of their sum?

Accepted Answer

$Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)$

Random Variables and Probability Distributions: Study Notes for Business Statistics

Study Guide - Practice Questions

Study Guide - Flashcards