Question 1

Which of the following statements best describes the difference between the two graphs shown above?

Accepted Answer

The left graph shows the probability distribution as individual points, while the right graph shows the same probabilities as bars, making it easier to compare the relative frequencies of each outcome.

Question 2

Suppose the random variable $$x$$ represents the number of heads observed when flipping two fair coins. Based on the graphs, what is the probability of observing exactly one head?

Accepted Answer

$$p(1) = \frac{1}{2}$$

Question 3

Given the probability distribution shown in the graphs, what is the expected value (mean) $$\mu of $$the random variable $$x$$?

Accepted Answer

$$\mu = 1$$

Question 4

If $$p(0) = \frac{1}{4}$$, $$p(1) = \frac{1}{2}$$, and $$p(2) = \frac{1}{4}$$, what is the variance $$\$$sigma^2$$ of x$?

Accepted Answer

$$\$$sigma^2$$ = 0.5$$

Question 5

Which of the following is a requirement for the probability distribution of a discrete random variable, as illustrated by the graphs?

Accepted Answer

$$\sum p(x) = 1$$

Question 6

If the random variable $$x in $$the graphs represents the number of successes in two independent trials with probability $$p = 0.5$$ for each success, which distribution does $$x$$ follow?

Accepted Answer

Binomial distribution with $$n = 2$$ and $$p = 0.5$$

Question 7

Suppose you observe $$x = 2 in $$this experiment. What is the probability of this outcome?

Accepted Answer

$$p(2) = \frac{1}{4}$$

Question 8

Which of the following best describes the type of random variable represented in the graphs?

Accepted Answer

Discrete random variable

Question 9

If the probability distribution is changed so that $$p(0) = 0.2$$, $$p(1) = 0.5$$, and $$p(2) = 0.3$$, does this still represent a valid probability distribution?

Accepted Answer

Yes, because $$0.2 + 0.5 + 0.3 = 1$$

Question 10

Which real-world scenario could be modeled by the probability distribution shown in the graphs?

Accepted Answer

Counting the number of heads when flipping two fair coins

Random Variables and Probability Distributions: Study Notes for Business Statistics

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