Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values are limited to distinct, separate outcomes.

Question 2

Given a discrete random variable $$X$$ with possible values $$x_1$$, $$x_2$$, ..., x_n$$ and corresponding probabilities P$$($$x_1$$), P($$x_2$$), ..., P(x_n)$$, which formula correctly calculates the expected value (mean) of X$?

Accepted Answer

$$\mu = \sum_{i=1}^n x_i \cdot P(x_i)$$

Question 3

A company gives every employee a $$5000 raise. If X$ is the random variable representing salary, what happens to the variance of X$?

Accepted Answer

The variance remains unchanged.

Question 4

If $$X is a $$random variable with mean $$\mu$$ and variance $$\$$sigma^2$$, what is the variance of aX$ for a constant a$?

Accepted Answer

$$a^2$$ \$$sigma^2$$

Question 5

Suppose you combine two independent random variables $$X$$ and $$Y. $$Which formula gives the variance of their sum?

Accepted Answer

$$Var(X+Y) = Var(X) + Var(Y)$$

Question 6

A binomial random variable $$X$$ counts the number of successes in $$n$$ independent Bernoulli trials with probability $$p of $$success. What is the expected value of $$X$$?

Accepted Answer

$$np$$

Question 7

Which condition must be met to use the Binomial model for Bernoulli trials?

Accepted Answer

Each trial must have only two possible outcomes and be independent.

Question 8

For a binomial random variable $$X$$ with $$n$$ trials and probability $$p of $$success, what is the formula for the standard deviation?

Accepted Answer

$$\sqrt{np(1-p)}$$

Question 9

You randomly select 25 emails, and 91% are spam. What is the expected number of real messages?

Accepted Answer

$$2.25$$

Question 10

A continuous random variable $$X$$ follows a uniform distribution on $$(a, b). $$What is the mean of $$X$$?

Accepted Answer

$$\frac{a+b}{2}$$

Question 11

For a uniform random variable $$X on$$ $$(a, b)$$, what is the variance?

Accepted Answer

$$\frac{(b-a)^2$}{12}$$

Question 12

If you wait for a train that comes every 10 minutes, and your wait time is uniformly distributed between 0 and 10 minutes, what is the probability you wait between 2 and 4 minutes?

Accepted Answer

$$\frac{4-2}{10-0} = 0.2$$

Question 13

When can the Normal model be used to approximate the Binomial model?

Accepted Answer

When both $$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 14

In the Normal approximation to the Binomial, what is the mean and variance?

Accepted Answer

Mean $$\mu = np$$, Variance $$\$$sigma^2$$ = np(1-p)$$

Question 15

A random variable $$X is $$shifted by adding a constant $$c. $$What happens to the standard deviation?

Accepted Answer

It remains unchanged.

Question 16

If you multiply a random variable $$X by a $$constant $$a$$, what happens to the standard deviation?

Accepted Answer

It becomes $$|a|$$ times the original standard deviation.

Question 17

Which of the following is a critical assumption for using the Binomial model?

Accepted Answer

The trials must be independent.

Question 18

If you randomly select more than 10% of a population, which assumption fails for the Binomial model?

Accepted Answer

Independence of trials.

Question 19

Which of the following statements is TRUE about combining variances of independent random variables?

Accepted Answer

Variances add, but standard deviations do not.

Question 20

In the context of probability models, what does the area under the density curve represent for a continuous random variable?

Accepted Answer

The probability of the random variable falling within a given interval.

Lw Probability Models for Random Variables: Discrete and Continuous Distributions

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