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

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