Q: Which formula correctly gives the variance of a discrete random variable $$X$$?

$$\text{Var}(X) = E[X^2$] - (E[X])^2$

Question 1

Which of the following best describes a random variable?

Accepted Answer

A variable that assigns a numerical value to each outcome of a random experiment.

Question 2

Suppose you toss three coins. Let $$X be $$the number of tails. What is $$P(X=2)$$?

Accepted Answer

$$\frac{3}{8}$$

Question 3

Which of the following is NOT a property of a probability distribution for a discrete random variable?

Accepted Answer

The expected value must always be an integer.

Question 4

A random variable $$X$$ has the following probability distribution: $$P(X=0)=0.25$$, $$P(X=1)=0.5$$, $$P(X=2)=0.25. $$What is the expected value $$E(X)$$?

Accepted Answer

$$1.0$$

Question 5

Which formula correctly gives the variance of a discrete random variable $$X$$?

Accepted Answer

$$	ext{Var}(X) = E[X^2$] - (E[X])^2$

Question 6

A fire insurance policy pays out $$200,000 if $$there is a fire, and $$0$$ otherwise. If the probability of a fire is $$0.0001$$, what is the expected payout $$E(X)$$?

Accepted Answer

$$20$$

Question 7

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

Accepted Answer

The number of emails received per hour.

Question 8

If $$X is a $$binomial random variable with $$n=10$$ and $$p=0.2$$, what is the expected value $$E(X)$$?

Accepted Answer

$$2$$

Question 9

A call center receives calls at a rate of $$3$$ per minute. What is the probability that exactly $$2$$ calls arrive in one minute? (Use the Poisson formula)

Accepted Answer

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

Question 10

Which of the following is a key assumption of the binomial distribution?

Accepted Answer

There are only two possible outcomes for each trial.

Question 11

Suppose 5% of manufactured resistors are defective. If you buy 20 resistors, what is the probability that none are defective? (Use the binomial formula)

Accepted Answer

$$P(X=0) = (0.95$$)^{20}$$

Question 12

Which formula gives the probability of $$r$$ successes in $$n$$ binomial trials with probability $$p of $$success?

Accepted Answer

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

Question 13

A Bernoulli random variable $$X$$ with $$p=0.01$$ models whether a resistor is defective. What is $$P(X=0)$$?

Accepted Answer

$$0.99$$

Question 14

Which of the following is NOT a discrete random variable?

Accepted Answer

The amount of rainfall in a day.

Question 15

If $$X is a $$Poisson random variable with mean $$\lambda=2$$, what is $$P(X=0)$$?

Accepted Answer

$$e^{-2}$$

Question 16

A box contains 20 tax return forms. $$X is $$the number of forms with errors. Which distribution best models $$X$$?

Accepted Answer

Binomial distribution

Question 17

Which formula gives the standard deviation of a discrete random variable $$X$$?

Accepted Answer

$$SD(X) = \sqrt{Var(X)}$$

Question 18

If the probability of a resistor being defective is $$0.01$$, and you test one resistor, what is the probability that it is not defective?

Accepted Answer

$$0.99$$

Question 19

Which of the following is a key difference between the binomial and Poisson distributions?

Accepted Answer

The binomial distribution models a fixed number of trials, while the Poisson distribution models events in a fixed interval.

Discrete Random Variables and Probability Distributions

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