Question 1

Which of the following is a requirement for a probability distribution?

Accepted Answer

The sum of all probabilities must equal 1.

Question 2

Given the frequency distribution: $$x = 1, 2, 3, 4, 5$$ and $$f = 24, 33, 42, 30, 21$$, what is the probability $$P(3)$$?

Accepted Answer

$$P(3) = \frac{42}{150} = 0.28$$

Question 3

A probability distribution is shown as $$P(x) = 0.216, 0.432, 0.288, 0.064$$ for $$x = 0, 1, 2, 3. $$What is the sum of the probabilities?

Accepted Answer

$$0.216 + 0.432 + 0.288 + 0.064 = 1$$

Question 4

Which formula represents the mean of a discrete random variable?

Accepted Answer

$$\mu = \sum xP(x)$$

Question 5

If a probability distribution has $$x = 1, 2, 3, 4, 5$$ and $$P(x) = 0.16, 0.22, 0.28, 0.20, 0.14$$, what is the mean $$\mu$$?

Accepted Answer

$$\mu = 1(0.16) + 2(0.22) + 3(0.28) + 4(0.20) + 5(0.14) = 2.94$$

Question 6

Which formula represents the variance of a discrete random variable?

Accepted Answer

$$\$$sigma^2$$ = \sum (x - \mu)^2$ P(x)$$

Question 7

A company tracks the number of sales new employees make each day during a 100-day probationary period. If the frequency distribution is $$x = 0, 1, 2, 3, 4, 5, 6, 7$$ and $$f = 16, 19, 15, 21, 9, 10, 8, 2$$, what is the probability that an employee makes exactly 3 sales in a day?

Accepted Answer

$$P(3) = \frac{21}{100} = 0.21$$

Question 8

Which of the following distributions is NOT a probability distribution?

Accepted Answer

$$P(x) = 0.3, 0.4, 0.5$$ for $$x = 1, 2, 3$$

Question 9

If the mean of a probability distribution is $$\mu = 2.9$$, what does this value represent?

Accepted Answer

The theoretical average of the random variable over many trials.

Question 10

Which formula represents the standard deviation of a discrete random variable?

Accepted Answer

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

Question 11

A meteorologist uses a tree diagram to determine the probability distribution for the number of days of rain in a three-day forecast. If the probability of rain each day is 0.4, what is the probability of exactly one day of rain?

Accepted Answer

$$P(1) = 0.432$$

Question 12

If a probability distribution is symmetric, what does this imply about the mean and the shape of the distribution?

Accepted Answer

The mean is at the center and the distribution is balanced on both sides.

Question 13

Which of the following statements about probability distributions is correct?

Accepted Answer

The sum of all probabilities must always equal 1.

Question 14

If $$P(x)$$ for $$x = 1, 2, 3, 4 is$$ $$0.3, 0.2, 0.4, 0.1$$, is this a probability distribution?

Accepted Answer

Yes, because all probabilities are between 0 and 1 and their sum is 1.

Question 15

In a probability distribution, what does the area of each bar in a histogram represent?

Accepted Answer

The probability of each outcome.

Question 16

If the variance of a probability distribution is $$\$$sigma^2$$ = 1.5$$, what is the standard deviation?

Accepted Answer

$$\sigma = \sqrt{1.5} \approx 1.225$$

Discrete Probability Distributions: Probability, Mean, Variance, and Standard Deviation

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