Q: A factory produces light bulbs with a 2% defect rate. If 10 bulbs are selected at random, what is the expected number of defective bulbs?

$$E(X) = np = 10 \times 0.02 = 0.2$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can only take integer values from a finite or countable set.

Question 2

Given the probability model for an insurance payout: $$x = 10,000$$ with $$P(X = 10,000) = 0.001$$, $$x = 5,000$$ with $$P(X = 5,000) = 0.002$$, and $$x = 0$$ with $$P(X = 0) = 0.997$$, what is the expected payout $$\mu$$?

Accepted Answer

$$\mu = 10,000(0.001) + 5,000(0.002) + 0(0.997) = 20$$

Question 3

Which formula correctly calculates the variance $$\$$sigma^2$$ of a discrete random variable X$ with mean $$\mu$$?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - \mu)^2$ P(x)$$

Question 4

A random variable $$X$$ has the following probability distribution: $$P(X=1) = 0.2$$, $$P(X=2) = 0.5$$, $$P(X=3) = 0.3. $$What is $$E(X)$$?

Accepted Answer

$$E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 2.1$$

Question 5

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

Accepted Answer

The probability of success changes from trial to trial.

Question 6

If $$X \sim 	ext{Binom}(n, p)$$, what is the formula for the probability of exactly $$x$$ successes?

Accepted Answer

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

Question 7

A factory produces light bulbs with a 2% defect rate. If 10 bulbs are selected at random, what is the expected number of defective bulbs?

Accepted Answer

$$E(X) = np = 10 	imes 0.02 = 0.2$$

Question 8

Which condition must be checked before using the Normal approximation to the Binomial distribution?

Accepted Answer

$$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 9

A Poisson random variable $$X$$ has mean $$\lambda = 3. $$What is the probability that $$X = 2$$?

Accepted Answer

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

Question 10

Which of the following statements about the Poisson distribution is TRUE?

Accepted Answer

The mean and variance are both equal to $$\lambda.$$

Question 11

A wildlife biologist expects to find a rare trait in 1 out of every 8 frogs. If he samples 150 frogs, what is the expected number and standard deviation of frogs with the trait?

Accepted Answer

Expected: $$18.75$$, Standard deviation: $$4.05$$

Question 12

Which scenario is best modeled by a Poisson distribution?

Accepted Answer

Counting the number of emails received per hour.

Question 13

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.8$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{20 	imes 0.8 	imes 0.2} = 1.8$$

Question 14

A pump fails on average once every 5000 hours. What is the probability of exactly two failures in 5000 hours using the Poisson model?

Accepted Answer

$$P(X=2) = \frac{1^2$ e$$^{-1}$$}{2!} \approx 0.1839$$

Question 15

Which of the following is a valid probability model for a discrete random variable?

Accepted Answer

All probabilities are between 0 and 1, and their sum is 1.

Question 16

If a random variable $$X$$ has $$E(X) = 5$$ and $$Var(X) = 4$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = \sqrt{4} = 2$$

Chapter 13: Random Variables – Discrete Probability Models, Binomial & Poisson Distributions

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