Question 1

Which of the following statements best describes the expected value of a discrete random variable $$x$$?

Accepted Answer

It is the theoretical mean outcome for infinitely many trials, calculated as $$\sum [x \cdot P(x)].$$

Question 2

A probability distribution for a discrete random variable must satisfy which of the following requirements?

Accepted Answer

The sum of all probabilities must equal 1, and each probability must be between 0 and 1 inclusive.

Question 3

Suppose a binomial experiment has $$n = 10$$ trials and the probability of success in each trial is $$p = 0.3. $$What is the mean number of successes?

Accepted Answer

Mean $$= n \cdot p = 10 \cdot 0.3 = 3$$

Question 4

Which scenario is best modeled by a Poisson probability distribution?

Accepted Answer

The number of patients arriving at an emergency room in one hour.

Question 5

A binomial probability distribution requires which of the following conditions?

Accepted Answer

The trials must be independent, and each trial must have exactly two outcomes.

Question 6

Given a binomial distribution with $$n = 20$$, $$p = 0.4$$, what is the standard deviation?

Accepted Answer

Standard deviation $$= \sqrt{n p q} = \sqrt{20 \cdot 0.4 \cdot 0.6} = \sqrt{4.8} \approx 2.19$$

Question 7

If the probability of getting 7 or more successes in a binomial experiment is $$0.03$$, how should this result be interpreted according to the rare event rule?

Accepted Answer

The outcome is significantly high and suggests the original assumption may not be correct.

Question 8

Which formula gives the probability of exactly $$x$$ successes in $$n$$ binomial trials?

Accepted Answer

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

Question 9

A Poisson distribution has a mean $$\mu = 4. $$What is the standard deviation?

Accepted Answer

Standard deviation $$= \sqrt{\mu} = \sqrt{4} = 2$$

Question 10

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

Accepted Answer

The random variable must be continuous.

Question 11

In a binomial experiment, if $$n = 100$$ and $$p = 0.05$$, can a Poisson distribution be used as an approximation? Why?

Accepted Answer

Yes, because $$n \geq 100$$ and $$np = 5 \leq 10.$$

Question 12

A random variable $$x is $$defined as the number of hurricanes in a year. Which type of random variable is $$x$$?

Accepted Answer

Discrete random variable

Question 13

If the mean number of patients arriving at an ER in one hour is $$\mu = 6$$, what is the probability that exactly 4 patients arrive in one hour? (Use the Poisson formula.)

Accepted Answer

$$P(4) = \frac{6^4$ \cdot e$$^{-6}$$}{4!}$$

Question 14

Which of the following best describes a continuous random variable?

Accepted Answer

A variable whose values are not countable and can take any value on a continuous scale.

Question 15

A binomial experiment has $$n = 15$$ trials and $$p = 0.2. $$What is the variance?

Accepted Answer

Variance $$= n p q = 15 \cdot 0.2 \cdot 0.8 = 2.4$$

Question 16

If the mean number of occurrences in a Poisson distribution is $$\mu = 3$$, what is the probability of zero occurrences?

Accepted Answer

$$P(0) = \frac{3^0$ \cdot e$$^{-3}$$}{0!} = e$$^{-3}$$

Question 17

Which of the following is a correct application of the range rule of thumb for identifying significantly high values?

Accepted Answer

A value is significantly high if it is greater than or equal to $$(\mu + 2\sigma).$$

Question 18

A probability distribution is given by $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the expected value?

Accepted Answer

Expected value $$= 0 \cdot 0.2 + 1 \cdot 0.5 + 2 \cdot 0.3 = 1.1$$

Question 19

Which of the following is a property of the Poisson distribution?

Accepted Answer

It is determined only by the mean $$\mu.$$

Probability Distributions: Binomial and Poisson Distributions in Introductory Statistics

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