Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of defective lightbulbs from a randomly chosen batch in a factory

Question 2

A probability distribution for a discrete random variable must satisfy which two criteria?

Accepted Answer

Each probability must be between 0 and 1, and the sum of all probabilities must be 1

Question 3

Given the probability distribution for the number of defective bulbs in a batch: 0 (0.20), 1 (0.30), 2 (0.25), 3 (0.15), 4 (0.07), 5 (0.03), what is the expected value?

Accepted Answer

$$\sum x \cdot P(x) = 0 \cdot 0.20 + 1 \cdot 0.30 + 2 \cdot 0.25 + 3 \cdot 0.15 + 4 \cdot 0.07 + 5 \cdot 0.03 = 1.58$$

Question 4

How do you calculate the variance of a discrete random variable?

Accepted Answer

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

Question 5

A binomial experiment is defined by which of the following characteristics?

Accepted Answer

A fixed number of trials, only two possible outcomes per trial, independent trials, and constant probability of success

Question 6

If a basketball player shoots until they make a basket, and the random variable is the number of shots taken, is this a binomial experiment?

Accepted Answer

No, there are not a fixed number of trials

Question 7

What is the binomial probability formula for finding the probability of exactly $$k$$ successes in $$n$$ trials?

Accepted Answer

$$P(X = k) = \binom{n}{k} p^k q$$^{n-k}$$

Question 8

A company expects 60% of customers to purchase a product after a demonstration. If 10 customers are selected, what is the expected number of purchases?

Accepted Answer

$$\mu = n \cdot p = 10 \cdot 0.6 = 6$$

Question 9

For a binomial distribution with $$n = 30$$ and $$p = 0.62$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{n p q} = \sqrt{30 \cdot 0.62 \cdot 0.38} \approx 2.67$$

Question 10

A gardener plants 8 seeds, each with a 65% probability of germinating. What is the probability that fewer than 4 seeds germinate?

Accepted Answer

$$P(X \u003c 4) = \sum_{x=0}^{3} \binom{8}{x} (0.65)^x (0.35$$)^{8-x}$$

Question 11

Which distribution is most appropriate for modeling the number of customers entering a bakery in a fixed 15-minute interval, given the average rate is 2 customers per 15 minutes?

Accepted Answer

Poisson distribution

Question 12

What is the Poisson probability formula for finding the probability of exactly $$x$$ occurrences in a fixed interval with mean rate $$\lambda$$?

Accepted Answer

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

Question 13

A quality control inspector finds an average of 0.5 defects per meter of fabric. What is the probability that 0 defects are found in a meter?

Accepted Answer

$$P(0) = \frac{0.5^0$ e$$^{-0.5}$$}{0!} = e$$^{-0.5}$$ \approx 0.6065$$

Question 14

When can the Poisson distribution be used to approximate the binomial distribution?

Accepted Answer

When $$n is $$large and $$p is $$small, such that $$n \geq 100$$ and $$np \leq 10$$

Question 15

A school raffle has 386 tickets, one for each student. Tickets are removed from the pool after being chosen and 5 tickets are drawn. Which distribution should be used?

Accepted Answer

Hypergeometric distribution

Question 16

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

Accepted Answer

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

Question 17

A shipment contains 100 units, 5 of which are defective. If 20 units are randomly selected without replacement, what is the probability that exactly 2 are defective?

Accepted Answer

$$P(2) = \frac{\binom{5}{2} \binom{95}{18}}{\binom{100}{20}}$$

Discrete Probability Distributions: Study Guide for Statistics Students

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