Question 1

Which of the following is NOT a requirement for a probability distribution of a discrete random variable $$X$$?

Accepted Answer

The random variable $$X$$ must take on only integer values.

Question 2

Suppose you toss two fair coins. Let $$X be $$the number of heads observed. What is the expected value (mean) $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = 1$$

Question 3

A discrete random variable $$X$$ has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the variance $$\$$sigma^2$$ of X$?

Accepted Answer

$$\$$sigma^2$$ = 0.41$$

Question 4

Which of the following scenarios is best modeled by a Poisson distribution?

Accepted Answer

The number of emails received per hour at a help desk.

Question 5

If a random variable $$X$$ has mean $$\mu = 10$$ and standard deviation $$\sigma = 2$$, what is the maximum usual value according to the Range Rule of Thumb?

Accepted Answer

$$\mu + 2\sigma = 14$$

Question 6

A probability distribution is considered well-defined if:

Accepted Answer

All probabilities are between 0 and 1, and their sum is 1.

Question 7

Suppose the probability that a randomly selected product is defective is $$0.02. In a $$sample of 100 products, what is the expected number of defective products?

Accepted Answer

2

Question 8

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

Accepted Answer

The number of cars passing through a toll booth in an hour.

Question 9

Given a probability distribution for $$X$$ with $$P(0) = 0.3$$, $$P(1) = 0.4$$, $$P(2) = 0.2$$, and $$P(3) = 0.1$$, what is $$P(X \geq 2)$$?

Accepted Answer

$$P(X \geq 2) = 0.3$$

Question 10

A roulette wheel has 38 slots. If you bet on a single number, the probability of winning is $$\frac{1}{38}. If $$the payout is $$35 to$$ $$1$$, what is the expected value of a $$\5$ bet?

Accepted Answer

$$-\4.74$

Question 11

If a random variable $$X$$ has the following probability distribution: $$P(0) = 0.5$$, $$P(1) = 0.3$$, $$P(2) = 0.2$$, what is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = 0.9$$

Question 12

Which of the following statements about the standard deviation $$\sigma of a $$discrete random variable $$X is $$TRUE?

Accepted Answer

$$\sigma is $$always non-negative.

Question 13

A table lists the proportion of unlicensed software in each country as: 0.91, 0.86, 0.82, 0.73, 0.27. Why does this NOT represent a probability distribution?

Accepted Answer

The values are not probabilities for a random variable, but proportions for different countries.

Question 14

If the mean number of bank failures per year is $$\lambda = 5.2$$, what is the standard deviation for the Poisson distribution?

Accepted Answer

$$\sqrt{5.2}$$

Question 15

Which of the following is a correct interpretation of the 'Rare Event Rule'?

Accepted Answer

If an observed event has a probability less than 0.05, it is considered unusual.

Question 16

A random variable $$X$$ has $$P(0) = 0.25$$, $$P(1) = 0.5$$, $$P(2) = 0.25. $$What is the standard deviation $$\sigma of$$ $$X$$?

Accepted Answer

$$\sigma = 0.7071$$

Question 17

In a Poisson distribution, which of the following statements is TRUE?

Accepted Answer

The mean and variance are equal.

Question 18

Suppose a probability distribution is given by $$P(0) = 0.3$$, $$P(1) = 0.4$$, $$P(2) = 0.2$$, $$P(3) = 0.1. $$What is the probability that $$X is $$less than 2?

Accepted Answer

$$P(X \u003c 2) = 0.7$$

Question 19

If a discrete random variable $$X$$ has possible values 0, 1, 2, and $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3$$, what is $$E[X^2$]$$?

Accepted Answer

$$E[X^2$] = 1.3$$

Discrete Probability Distributions: Concepts, Calculations, and Applications

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