Q: Given the probability distribution for the number of kids per household: 0 kids (0.15), 1 kid (0.60), 2 kids (0.25), what is the expected value?

$$\ 0 \times 0.15 + 1 \times 0.60 + 2 \times 0.25 = 1.10 \ $$

Q: 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?

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

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 exactly 1

Question 3

Given the probability distribution for the number of kids per household: 0 kids (0.15), 1 kid (0.60), 2 kids (0.25), what is the expected value?

Accepted Answer

$$\ 0 	imes 0.15 + 1 	imes 0.60 + 2 	imes 0.25 = 1.10 \ $$

Question 4

How do you calculate the variance $$\ \$$sigma^2$$ \  of a $$discrete random variable $$\ X \ $$ with probability distribution?

Accepted Answer

$$\ \sum [(x - \mu)^2$ \cdot P(x)] \ $$

Question 5

A binomial experiment must satisfy which of the following conditions?

Accepted Answer

There must be a fixed number of independent trials, each with the same probability of success

Question 6

A basketball player shoots until they make a basket. Is this a binomial experiment?

Accepted Answer

No, because there are not a fixed number of trials

Question 7

What is the binomial probability formula for exactly $$\ x \ $$ successes in $$\ n \ $$ trials with probability $$\ p \  of $$success?

Accepted Answer

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

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

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

Question 9

For a binomial distribution, what is the formula for the standard deviation $$\ \sigma \ $$?

Accepted Answer

$$\ \sigma = \sqrt{n \cdot p \cdot q} \ $$

Question 10

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

Accepted Answer

$$\ \sum_{x=0}^{3} \binom{8}{x} \cdot 0.65^x \cdot 0.35$$^{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 exactly $$\ x \ $$ occurrences given mean rate $$\ \lambda \ $$?

Accepted Answer

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

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

Question 14

A school raffle has 386 tickets, one per student. Tickets are removed after being chosen and 5 tickets are drawn. Which distribution should be used to model the number of winners from a particular class?

Accepted Answer

Hypergeometric distribution

Question 15

What is the formula for the probability of $$\ x \ $$ successes in a hypergeometric distribution, given $$\ r \ $$ successes in a population of $$\ N \ $$ items and $$\ n \ $$ draws?

Accepted Answer

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

Question 16

A company produces 1,000 light bulbs daily with a defect rate of 5%. What is the expected number of defective bulbs?

Accepted Answer

$$\ \mu = n \cdot p = 1000 	imes 0.05 = 50 \ $$

Question 17

A customer service call center receives an average of 12 calls per hour. What is the expected number of calls in a 30-minute period?

Accepted Answer

$$\ \lambda = 12 	imes 0.5 = 6 \ $$

Discrete Probability Distributions: Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard