Question 1

Which of the following is a requirement for a table to represent a probability distribution for a discrete random variable $$x$$?

Accepted Answer

Each probability must be between 0 and 1 inclusive, and the sum of all probabilities must equal 1.

Question 2

Given the following probability distribution for the number of households with a printer in a group of four: $$x = 0, 1, 2, 3, 4$$ with $$P(x) = 0.029, 0.142, 0.334, 0.358, 0.137$$ respectively, what is the mean ($$\mu) of$$ $$x$$?

Accepted Answer

$$\mu = 2.4$$

Question 3

For the same probability distribution, what is the standard deviation ($$\sigma) of$$ $$x$$? (Round to one decimal place.)

Accepted Answer

$$\sigma = 0.9$$

Question 4

Suppose a probability distribution table for a random variable $$x$$ contains a probability value of $$1.2. $$Which requirement for a probability distribution is violated?

Accepted Answer

Not every probability is between 0 and 1 inclusive.

Question 5

If the sum of all probabilities in a distribution is $$0.95$$, what does this indicate?

Accepted Answer

The sum of all probabilities is not equal to 1.

Question 6

A group of four households is surveyed for printer ownership. If $$P(x=3) = 0.358$$, what does this probability represent?

Accepted Answer

The probability that exactly three households in a group of four have a printer.

Question 7

Which formula is used to calculate the mean ($$\mu) of a $$discrete probability distribution?

Accepted Answer

$$\mu = \sum x \cdot P(x)$$

Question 8

Which formula is used to calculate the standard deviation ($$\sigma) of a $$discrete probability distribution?

Accepted Answer

$$\sigma = \sqrt{\sum (x - \mu)^2$ \cdot P(x)}$$

Question 9

If a probability distribution is given by $$x = 0, 1, 2$$ with $$P(x) = 0.2, 0.5, 0.3$$, what is the expected value of $$x$$?

Accepted Answer

$$1.1$$

Question 10

A researcher claims that the random variable $$x in a $$probability distribution must always be categorical. Is this statement correct?

Accepted Answer

No, because $$x$$ must be numerical for a probability distribution.

Question 11

In a real-world scenario, why is it important that the sum of all probabilities in a distribution equals 1?

Accepted Answer

It ensures that all possible outcomes are accounted for.

Question 12

Suppose you are given a probability distribution for the number of defective items in a sample of 4, with $$P(x)$$ values all between 0 and 1, and their sum is 1. What can you conclude?

Accepted Answer

The table represents a valid probability distribution.

Question 13

If the probability that none of four households has a printer is $$0.029$$, what is the probability that at least one household has a printer?

Accepted Answer

$$0.971$$

Question 14

A probability distribution for $$x ($$number of printers in a group of four households) is given. If you want to find the probability that at most two households have a printer, which calculation is correct?

Accepted Answer

$$P(x=0) + P(x=1) + P(x=2)$$

Probability Distribution, Mean, and Standard Deviation for a Discrete Random Variable

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