Q: A roulette wheel has 38 numbers (18 red, 18 black, 2 green). If you bet $5 on red, what is the expected value of your winnings?

$$E = 5 \times \frac{18}{38} + (-5) \times \frac{20}{38} = -0.26$$

Q: Given a binomial distribution with $$n=10$$ and $$p=0.55$$, what is the mean?

$$\mu = np = 10 \times 0.55 = 5.5$$

Q: If the mean voter turnout in a sample of 1002 people is 61%, what is the expected number of voters?

$$E = np = 1002 \times 0.61 = 611.22$$

Q: A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E(X)$$?

$$E(X) = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Question 1

Which formula correctly represents the standard deviation of a binomial distribution?

Accepted Answer

$$\sigma = \sqrt{npq}$$

Question 2

A random variable $$X is $$defined as the number of heads in three flips of a fair coin. Which of the following is the correct probability distribution for $$X$$?

Accepted Answer

$$P(X=0)=0.125,\ P(X=1)=0.375,\ P(X=2)=0.375,\ P(X=3)=0.125$$

Question 3

Which of the following is NOT a requirement for a probability distribution?

Accepted Answer

The sum of all probabilities must be greater than 1

Question 4

A roulette wheel has 38 numbers (18 red, 18 black, 2 green). If you bet $5 on red, what is the expected value of your winnings?

Accepted Answer

$$E = 5 	imes \frac{18}{38} + (-5) 	imes \frac{20}{38} = -0.26$$

Question 5

Which of the following best describes a continuous random variable?

Accepted Answer

It can take any value within an interval

Question 6

Given a binomial distribution with $$n=10$$ and $$p=0.55$$, what is the mean?

Accepted Answer

$$\mu = np = 10 	imes 0.55 = 5.5$$

Question 7

Which of the following scenarios is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective items in a fixed sample of 20 products

Question 8

The graph shown is symmetric and bell-shaped. Which distribution does it represent?

Accepted Answer

Normal distribution

Question 9

If the mean voter turnout in a sample of 1002 people is 61%, what is the expected number of voters?

Accepted Answer

$$E = np = 1002 	imes 0.61 = 611.22$$

Question 10

A bone density test result is normally distributed with $$\mu=0$$ and $$\sigma=1. $$What is the probability that a randomly selected person has a z-score less than 1.27?

Accepted Answer

0.898

Question 11

Which of the following is the correct formula for the variance of a binomial distribution?

Accepted Answer

$$\$$sigma^2$$ = npq$$

Question 12

In a normal distribution with $$\mu=68.6$$ and $$\sigma=2.8$$, what percentage of men are between 62 and 77 inches tall?

Accepted Answer

99.1%

Question 13

Which of the following is NOT a property of a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial

Question 14

A z-score separates the bottom 2.5% from the top 97.5% in a standard normal distribution. What is the value of this z-score?

Accepted Answer

$$z = -1.96$$

Question 15

If $$P(X \u003e x) = 0.11$$ for a normal distribution, which of the following best describes the shaded region in the graph?

Accepted Answer

The area to the right of $$x is 0.11$$

Question 16

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.2$$, $$P(X=1)=0.5$$, $$P(X=2)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Probability Distributions: Discrete, Binomial, and Normal Distributions

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