Q: Suppose $$X \sim \text{Bin}(n=10, p=0.3). $$What is the standard deviation of $$X$$?

$$\sqrt{10 \times 0.3 \times 0.7} = 1.449$$

Q: A life insurance company pays out $$10,000 if a $$customer dies and collects $$250 if $$the customer lives. If the probability of death is $$0.001$$, what is the expected gain for the company per policy?

$$E = 250 \times 0.999 - 10,000 \times 0.001 = 240$$

Question 1

Which of the following best defines a random variable?

Accepted Answer

A function that assigns a numeric value to each outcome of a random experiment.

Question 2

Suppose you roll a fair six-sided die. Let $$X be $$the number that appears on the die. What is the probability distribution for $$X$$?

Accepted Answer

$$P(X = x) = \frac{1}{6}$$ for $$x = 1, 2, 3, 4, 5, 6$$

Question 3

Which of the following is NOT a requirement for a valid probability distribution for a discrete random variable $$X$$?

Accepted Answer

The mean of the distribution must be zero.

Question 4

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

Accepted Answer

$$E(X) = 1.2$$

Question 5

A probability distribution for a random variable $$X is $$given by $$P(X=1)=0.4$$, $$P(X=2)=0.3$$, $$P(X=3)=0.2$$, $$P(X=4)=0.1. Is $$this a valid probability distribution?

Accepted Answer

Yes, because all probabilities are positive and sum to 1.

Question 6

Suppose a geneticist records the number of mutated alleles in a plant, with $$X$$ taking values $$1, 2, 3$$ and $$P(X=1)=0.5$$, $$P(X=2)=0.3$$, $$P(X=3)=0.2. $$What is the variance of $$X$$?

Accepted Answer

$$\$$sigma^2$$ = 0.7$$

Question 7

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

Accepted Answer

Counting the number of heads in 10 flips of a fair coin.

Question 8

A binomial random variable $$X$$ has parameters $$n=8$$ and $$p=0.25. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 2$$

Question 9

Suppose $$X \sim 	ext{Bin}(n=10, p=0.3). $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{10 	imes 0.3 	imes 0.7} = 1.449$$

Question 10

A life insurance company pays out $$10,000 if a $$customer dies and collects $$250 if $$the customer lives. If the probability of death is $$0.001$$, what is the expected gain for the company per policy?

Accepted Answer

$$E = 250 	imes 0.999 - 10,000 	imes 0.001 = 240$$

Question 11

Which of the following is the formula for the probability of exactly $$k$$ successes in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

$$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

Question 12

A probability distribution is given by $$P(X=0)=0.2$$, $$P(X=1)=0.3$$, $$P(X=2)=0.6. Is $$this a valid probability distribution?

Accepted Answer

No, because the probabilities sum to more than 1.

Question 13

Suppose you are testing a new vaccine and the probability of a side effect is $$0.2$$ for each participant. If you test 10 participants, what is the expected number of side effects?

Accepted Answer

$$E(X) = 10 	imes 0.2 = 2$$

Question 14

Which of the following statements about the mean and standard deviation of a discrete probability distribution is correct?

Accepted Answer

The mean is the weighted average of the possible values, and the standard deviation measures the spread around the mean.

Question 15

A random variable $$X$$ has $$P(X=0)=0.3$$, $$P(X=1)=0.4$$, $$P(X=2)=0.3. $$What is the variance of $$X$$?

Accepted Answer

$$\$$sigma^2$$ = 0.6$$

Question 16

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of defective items in a batch of 100.

Question 17

Suppose $$X is a $$binomial random variable with $$n=12$$ and $$p=0.15. $$What is the probability of exactly 2 successes?

Accepted Answer

$$P(X=2) = \binom{12}{2} (0.15)^2$ (0.85$$)^{10}$$

Question 18

Which of the following best describes the probability mass function (pmf) of a discrete random variable?

Accepted Answer

A function that gives the probability that a discrete random variable is exactly equal to some value.

Discrete Probability Distributions: Random Variables, Probability Distributions, and the Binomial Distribution

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