Question 1

Which of the following best describes the random variable represented by the first image (number line with only points at 0, 1, 2, 3, 4)?

Accepted Answer

A discrete random variable because it can only take specific, countable values.

Question 2

Which of the following best describes the random variable represented by the second image (number line shaded continuously from 0 to 4)?

Accepted Answer

A continuous random variable because it can take any value within the interval.

Question 3

Based on the tree diagram in the third and fourth images, what is the probability of getting exactly 2 successes in 4 independent trials if the probability of success in each trial is $$p$$?

Accepted Answer

$$6 \cdot p^2$ \cdot (1-p)^2$

Question 4

According to Table 9, which mathematical symbol corresponds to the phrase 'no more than'?

Accepted Answer

$$\leq$$

Question 5

Which of the following is NOT a requirement for a binomial experiment?

Accepted Answer

The probability of success changes from trial to trial.

Question 6

Given a binomial experiment with $$n=4$$ and $$p=0.7$$, what is the probability of exactly 3 successes?

Accepted Answer

$$4 \choose 3$$ $$\cdot 0.7^3$ \cdot 0.3$$

Question 7

If $$X is a $$discrete random variable with the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3$$, what is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = 0 \cdot 0.2 + 1 \cdot 0.5 + 2 \cdot 0.3$$

Question 8

Which of the following statements about the mean of a discrete random variable is TRUE?

Accepted Answer

It is the long-run average value of the random variable over many repetitions.

Question 9

Suppose a random variable $$X$$ has mean $$\mu = 2$$ and standard deviation $$\sigma = 1. $$According to the empirical rule for a binomial distribution, which interval would contain most of the observed values?

Accepted Answer

$$[1, 3]$$

Question 10

If a discrete random variable $$X$$ has the following probability distribution: $$P(-1) = 0.5$$, $$P(2) = 0.5$$, what is the standard deviation $$\sigma of$$ $$X$$?

Accepted Answer

$$\sigma = \sqrt{0.5 \cdot (-1 - 0.5)^2$ + 0.5 \cdot (2 - 0.5)^2$}$$, where $$\mu = 0.5$$

Question 11

A casino game pays $$-1$$ with probability $$0.6$$, $$2$$ with probability $$0.3$$, and $$5$$ with probability $$0.1. $$What is the expected value $$E(X) of a $$single play?

Accepted Answer

$$E(X) = -1 \cdot 0.6 + 2 \cdot 0.3 + 5 \cdot 0.1$$

Question 12

Which of the following scenarios describes a continuous random variable?

Accepted Answer

The mass of a randomly selected apple.

Question 13

A binomial experiment is performed with $$n=10$$ and $$p=0.4. $$What is the mean $$\mu$$ and standard deviation $$\sigma of $$the number of successes?

Accepted Answer

$$\mu = 10 \cdot 0.4$$, $$\sigma = \sqrt{10 \cdot 0.4 \cdot 0.6}$$

Question 14

Which phrase from Table 9 corresponds to the probability $$P(X \u003e 3)$$?

Accepted Answer

'more than 3'

Question 15

A researcher records the number of defective items in each box of 20 produced by a factory. What type of random variable is this?

Accepted Answer

Discrete

Discrete and Binomial Probability Distributions: Study Guide

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