Q: A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective bulbs in a batch is: 0 (0.29), 1 (0.40), 2 (0.21), 3 (0.07), 4 (0.03). What is the expected number of defective bulbs per batch?

$$0 \times 0.29 + 1 \times 0.40 + 2 \times 0.21 + 3 \times 0.07 + 4 \times 0.03 = 1.14$$

Q: In a lottery, the profits and probabilities are: $$-1$$ (0.99), $$1,000$$ (0.009), $$1,000,000$$ (0.001). What is the mean profit?

$$(-1) \times 0.99 + 1,000 \times 0.009 + 1,000,000 \times 0.001 = 1,000.01$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that represents a countable number of outcomes.

Question 2

Given the probability distribution for the number of sodas a person drinks per day: 0 sodas (0.10), 1 soda (0.33), 2 sodas (0.25), 3 sodas (0.18), 4 sodas (0.14), what is the probability that a person drinks at least 2 sodas per day?

Accepted Answer

$$0.25 + 0.18 + 0.14 = 0.57$$

Question 3

Which of the following is NOT a discrete random variable?

Accepted Answer

The time it takes for a randomly selected runner to complete a 5K race.

Question 4

A probability distribution must satisfy which of the following conditions?

Accepted Answer

The sum of all probabilities must equal 1.

Question 5

If $$X is a $$discrete random variable with probability distribution $$P(X)$$, what is the formula for the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \sum X \cdot P(X)$$

Question 6

A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective bulbs in a batch is: 0 (0.29), 1 (0.40), 2 (0.21), 3 (0.07), 4 (0.03). What is the expected number of defective bulbs per batch?

Accepted Answer

$$0 	imes 0.29 + 1 	imes 0.40 + 2 	imes 0.21 + 3 	imes 0.07 + 4 	imes 0.03 = 1.14$$

Question 7

Which formula is used to calculate the variance $$\$$sigma^2$$ of a discrete random variable X$?

Accepted Answer

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

Question 8

A company tracks the number of complaints they receive per month. The probability distribution is: 0 complaints (0.20), 1 complaint (0.30), 2 complaints (0.25), 3 complaints (0.15), 4 complaints (0.10). What is the variance of the number of complaints?

Accepted Answer

$$\$$sigma^2$$ = \sum (X - \mu)^2$ P(X)$$, \mu = 1.65, \$$sigma^2 = 1.4275$

Question 9

In a lottery, the profits and probabilities are: $$-1$$ (0.99), $$1,000$$ (0.009), $$1,000,000$$ (0.001). What is the mean profit?

Accepted Answer

$$(-1) 	imes 0.99 + 1,000 	imes 0.009 + 1,000,000 	imes 0.001 = 1,000.01$$

Question 10

Which of the following is a property of a probability distribution for a discrete random variable?

Accepted Answer

Each probability must be between 0 and 1, inclusive.

Question 11

If $$E(X) = 2.5$$ and $$\$$sigma^2$$ = 1.25$$ for a discrete random variable $$X$$, what is the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{1.25} = 1.12$$

Question 12

A random survey asks how many sodas people drink per day. What is the probability that a person drinks between 1 and 4 sodas per day, given the probabilities: 1 (0.33), 2 (0.25), 3 (0.18), 4 (0.14)?

Accepted Answer

$$0.33 + 0.25 + 0.18 + 0.14 = 0.90$$

Question 13

Which scenario best illustrates the use of a discrete random variable in a real-world context?

Accepted Answer

Counting the number of cars passing through a toll booth each hour.

Question 14

If $$P(X=2) = 0.25$$ and $$P(X=3) = 0.18$$, what is the probability that $$X is $$either 2 or 3?

Accepted Answer

$$0.25 + 0.18 = 0.43$$

Question 15

A probability distribution is given by $$P(X=0)=0.10$$, $$P(X=1)=0.33$$, $$P(X=2)=0.25$$, $$P(X=3)=0.18$$, $$P(X=4)=0.14. $$What is the sum of all probabilities?

Accepted Answer

$$0.10 + 0.33 + 0.25 + 0.18 + 0.14 = 1.00$$

Question 16

Which formula would you use to find the standard deviation $$\sigma of a $$discrete random variable $$X$$?

Accepted Answer

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

Question 17

A student wants to know the probability of breaking even in a lottery where the profits and probabilities are: $$-1$$ (0.99), $$1,000$$ (0.009), $$1,000,000$$ (0.001). What is the probability of at least breaking even?

Accepted Answer

$$0.009 + 0.001 = 0.01$$

Discrete Random Variables and Probability Distributions: Study Notes

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