Question 1

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

Accepted Answer

The sum of all probabilities must equal 1: $$\sum P(x) = 1$$

Question 2

A random variable $$X is $$defined as the number of heads in three tosses of a fair coin. What is the probability that $$X = 2$$?

Accepted Answer

$$P(X=2) = \dfrac{3}{8}$$

Question 3

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of defective light bulbs in a box

Question 4

Given the probability distribution: $$x: 0, 1, 2$$; $$P(x): 0.2, 0.5, 0.3$$, what is the expected value $$E[X]$$?

Accepted Answer

$$E[X] = 1.1$$

Question 5

Which of the following tables does NOT represent a valid probability distribution?

Accepted Answer

$$x: 1, 2, 3$$;\ P(x): 0.3, 0.4, 0.4$

Question 6

A lottery game costs $$1 to play. You win 100 if $$you guess the correct two-digit number (from 00 to 99). What is your expected net profit per game?

Accepted Answer

$$E = -\0.99$

Question 7

If $$Y = 3X + 2$$ and $$E[X] = 4$$, what is $$E[Y]$$?

Accepted Answer

$$E[Y] = 14$$

Question 8

Which of the following statements about variance is correct for a probability distribution?

Accepted Answer

Variance is calculated as $$\$$sigma^2$$ = \sum [x^2$ P(x)] - \mu^2$

Question 9

A table lists the proportion of unlicensed software in several countries: United States (0.17), China (0.70), India (0.58), Russia (0.64). Why does this NOT represent a probability distribution?

Accepted Answer

The values are not numerical random variables and the probabilities do not sum to 1

Question 10

Suppose $$X is a $$discrete random variable with $$E[X] = 2$$ and $$Var(X) = 3. If$$ $$Y = -2X$$, what is $$Var(Y)$$?

Accepted Answer

$$Var(Y) = 12$$

Question 11

Which of the following best describes a continuous random variable?

Accepted Answer

A variable that can take any value within an interval, such as body temperature

Question 12

Given the probability distribution: $$x: 0, 1, 2$$; $$P(x): 0.25, 0.5, 0.25$$, what is the standard deviation?

Accepted Answer

$$\sigma = 0.7$$

Question 13

If $$X is a $$discrete random variable and $$a$$ and $$b$$ are constants, which formula gives the variance of $$Y = aX + b$$?

Accepted Answer

$$Var(Y) = a^2$ Var(X)$$

Question 14

A probability distribution is given by $$x: 1, 2, 3$$; $$P(x): 0.2, 0.5, 0.3. $$What is the variance?

Accepted Answer

$$Var(X) = 0.41$$

Question 15

Which of the following is NOT a requirement for a probability distribution?

Accepted Answer

The random variable must be categorical

Question 16

If $$E[X] = 5$$ and $$Var(X) = 4$$, what is the standard deviation of $$X$$?

Accepted Answer

$$\sigma = 2$$

Question 17

A student claims that the sum of the probabilities in a probability distribution can be less than 1 if there are only a few possible outcomes. Is this correct?

Accepted Answer

No, the sum must always be exactly 1

Question 18

Suppose $$X is $$the number of heads in two tosses of a fair coin. What is the probability distribution for $$X$$?

Accepted Answer

$$P(0) = 0.25,\ P(1) = 0.5,\ P(2) = 0.25$$

5.1:Discrete Probability Distributions: Concepts, Properties, and Examples

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