A probability distribution specifies the probabilities of the possible outcomes of a random variable.

If you toss a coin 3 times, the possible outcomes are as follows (where H means heads and T means tails): TTT, TTH, THT, HTT, THH, HTH, HHT, HHH.

In total, there are 8 possible outcomes. Of these:

• Only 1 (TTT) has 0 heads occurring.
• Three (TTH, THT and HTT) have 1 heads occurring.
• Three (THH, HTH and HHT) have 2 heads occurring.
• One (HHH) has 3 heads occurring.

Thus, if x = number of heads in 3 tosses of a coin, then x = 0, 1, 2 or 3.

Now, the respective probabilities are 1/8, 3/8, 3/8 and 1/8, as you have just seen. So:

p(0) = p(0 Heads) = 1/8
p(1) = p(1 Head) = 3/8
p(2) = p(2 Heads) = 3/8
p(3) = p(3 Heads) = 1/8

This is a probability distribution; it records probabilities for each possible outcome of the random variable.

Discrete Probability Distribution

A table, graph or rule that associates a probability P(X=x_i) with each possible value x_i that the discrete random variable X can assume is called a discrete probability distribution. It is a theoretical model for the relative frequency distribution of a population.

A random variable is a quantity whose future outcomes are uncertain. Depending on the characteristics of the random variable, a probability distribution may be either discrete or continuous.

A discrete variable is one that cannot take on all values within the limits of the variable. It can assume only a countable number of possible values. For example, responses to a five-point rating scale can only take on the values 1, 2, 3, 4, and 5. The variable cannot have the value 1.7. The variable "number of correct answers on a 100-point multiple-choice test" is also a discrete variable since it is not possible to get 54.12 problems correct. The number of movies you will see this year, the number of trades a broker will perform next month, and the number of securities in a portfolio are all examples of discrete variables.

A continuous variable is one within the limits of variable ranges for which any value is possible. The number of possible values cannot be counted, and, as you will see later, each individual value has zero probability associated with it. For example, the variable "time to solve an anagram problem" is continuous since it could take 2 minutes or 2.13 minutes, etc., to finish a problem. A variable such as a person's height can take on any value as well. The rate of return on an asset is also a continuous random variable since the exact value of the rate of return depends on the desired number of decimal spaces.

Statistics computed from discrete variables are continuous. The mean on a five-point scale could be 3.117 even though 3.117 is not possible for an individual score.

For any random variable, it is necessary to know two things:

• the list of all possible values that the random variable can take on.
• the probability of each value occurring.

These give a probability distribution. The first item on the list is called the range.

With regard to the range of possible outcomes of a specified random variable:

• Sometimes the possible values of a random variable have both lower and upper bounds. For example, there are three possible values of the number of heads showing face-up on two tosses of a coin: 0, 1, and 2. Therefore, the lower bound is 0 and the upper bound is 2.

• Sometimes the lower bound exists, but the upper bound does not. For example, the lower bound of the price of a stock is 0, since it cannot fall below 0. However, there is no upper bound on the price (at least theoretically).

• Sometimes the upper bound exists, but the lower bound does not. Consider the profit or loss of the seller of a call option. Suppose the buyer pays the seller $2 to buy a call option, which gives the buyer the right to buy a stock at $10 by the end of 2006. The maximum profit the seller can make is $2, but the maximum loss the seller may incur is unlimited since there is no upper bound on the possible values of stock prices.

• In other cases, neither bound is obvious. Consider the profit or loss of a big company. In a good year, profits could be as high as dozens of billions of dollars, losses could be equivalent in a very bad year.

Practice Question 1

Suppose we have a loaded die that gives the outcomes 1 through 6 according to the following probability distribution.

Note that for this die all outcomes are not equally likely, as they would be if this were a fair die. If this die is rolled 6000 times, the number of times we get a 2 or a 3 should be about ______.

A. 1000
B. 2000
C. 3000

Correct Answer: C

Because the die is not fair, we need to use the probability distribution to determine the probabilities of the different outcomes. For this die, the probability of a 2 or 3 is 0.2 + 0.3 = 0.5. Therefore, in 6000 rolls we should get a 2 or a 3 about 0.5 x 6000 = 3000 times.

Practice Question 2

A die is altered to have faces showing 1, 2, 2, 3, 3, 3. If the die is rolled, p(2) = ______.

A. 1/6
B. 2/6
C. 1/2

Correct Answer: B

The probability of an event is the ratio of success to total. Here, we have a total of 6 different outcomes, of which 2 are a success (the 2 faces containing a 2). So, p(2) = 2/6.

Practice Question 3

For the bar graph shown below, p(10) = ______.

A. 1
B. 2/12
C. 1/12

Correct Answer: B

Reading across to 10 on the horizontal axis, we see that the height of the bar is 2. The vertical axis is in 1/12s, so p(10) = 2/12.

Practice Question 4

For the probability distribution shown below, P(x < 4) = ______.

A. 3/36
B. 30/36
C. 6/36

Correct Answer: A

P(x < 4) = p(2) + p(3). Adding these values, we get P(x < 4) = 3/36

Practice Question 5

For the probability distribution shown below ______.

A. P(x < 5) = 95%
B. The expected value of x is 2.2
C. P(x > 2) = 40%

Correct Answer: B

Reading the chart, we get: P(x < 5) = 1 - P(x > 5) = 1 - [p(5) + p(6)] = 1 - [0.05 + 0.05]= 90%.
P(x > 2) = 1 - P(x < 2) = 1 - [p(0)+p(1)+p(2)] = 1 - [0.1 + 0.3 + 0.25] = 0.35 = 35%
Expected value = summation of x times p(x). Using the calculator, we get 2.2.

Practice Question 6

Determine whether each random variable, x, is discrete or continuous.

I. The number of quarterbacks currently playing in the NFL
II. The current ages of the quarterbacks

Correct Answer: discrete; continuous

Practice Question 7

I select two cards from a deck of 52 cards and observe the color of each (26 cards in the deck are red and 26 are black). Which of the following is an appropriate sample space S for the possible outcomes?

A. S = {red, black}
B. S = {(red, red), (red, black), (black, red), (black, black)}, where, for example, (red, red) stands for the event "the first card is red and the second card is red."
C. S = {0, 1, 2}

Correct Answer: B

Each time I select two cards, each such selection results in either (red, red), (red, black), (black, red) or (black, black).

Practice Question 8

What is the principal distinction between a discrete and continuous random variable? Give an example of each.

Correct Answer: A discrete random variable can assume a countable number of values, while a continuous random variable can assume an infinite number of values.

Examples of a discrete random variable:

• the number of heads obtained when a coin is flipped three times.
• the number that turns up when a die is rolled.
• the number of people waiting in line at a movie theater

Examples of a continuous random variable:

• the height of a person.
• the amount of rainfall.
• time required to run a mile.

Practice Question 9

A continuous variable ______.

A. may take on only integer values (e.g., 0, 1, 2, ...)
B. may take on only a finite number of different values
C. may take on an infinite number of values
D. must be any non-negative real number

Correct Answer: C

Practice Question 10

Which of the following statements is incorrect?

A. The interest rate is a continuous variable.
B. Probability density function always refers to a continuous random variable.
C. The quoted stock price on the New York Stock Exchange is always a continuous random variable.

Correct Answer: C

This statement is incorrect because stock price is a discrete variable.

Practice Question 11

A medical center wants to estimate the average number of patients that visit their facility monthly. They think that this number varies from 500 to 1400. Should they opt to treat the number of monthly patients as a continuous or a discrete random variable?

A. discrete
B. continuous
C. either one

Correct Answer: B

Practice Question 12

The NASDAQ Composite Index had 2,407.650 points on January 5, 2011 and 1,987.260 points on December 28, 2011. It peaked on January 5, 2011 at 2,781.300, and it reached its lowest point in 2011 on September 21 at 1,423.190. If a person is interested in the performance of this index during 2011, what should the value be for the upper range of possible outcomes?

A. 1,423.190
B. 1,987.260
C. 2,407.650
D. 2,781.300

Correct Answer: D

The index peaked at 2,781.300 in 2011, so that should be the value of the upper side of the range.

Practice Question 13

A probability function is a rule of correspondence or equation that ______.

A. assigns values of x to the events of a probability experiment
B. assigns probabilities to the various values of x
C. defines the variability in an experiment

Correct Answer: B

Practice Question 14

Which of the following statements is incorrect?

A. A probability distribution always specifies the probabilities of the possible outcomes for a random variable.
B. Only lower-case letters can be used to denote random variables.
C. The outcomes of a continuous variable are infinite.

Correct Answer: B

We normally use capital letters to denote random variables.

Practice Question 15

Which of the following statements is not false?

A. A random variable can take on specific values with certainty.
B. The color of a car cannot be a random variable because it is not a random number.
C. A random variable is a function which associates an outcome with a real number.

Correct Answer: C

A random variable can take on specific values but with uncertainty. The color of a car can be a random variable; each color considered is an outcome which can be associated with a probability (real number).

Practice Question 16

Suppose you have a discrete probability function such that P(X=5) = 10%, P(X=6) = 20%, P(X=7) = 30%, P(X=8) = 20%, P(X=9) = 20%. Find F(8).

A. 20%
B. 60%
C. 80%

Correct Answer: C

F(8) is the probability that the function takes on values less than or equal to 8. So, F(8) = p(5) + p(6) + p(7) + p(8) = 0.10 + 0.20 + 0.30 + 0.20 = 0.80 = 80%.

Practice Question 17

Which of the following statements is true for random variables?

A. A random variable assumes numerical values associated with the random outcomes of an experiment; only one value can be assigned to each sample point.
B. A random variable can only assume discrete values.
C. A random variable assumes numerical values determined by a random number generator.

Correct Answer: A

Practice Question 18

The two general types of random variables are ______.

A. discrete and continuous
B. binomial and Poisson
C. Poisson and continuous

Correct Answer: A

Practice Question 19

An investor wants to study movements of the interest rate. He will collect data every quarter and register whether the interest rate decreased, increased, or remained the same in each quarter. Should this investor opt to treat movements of the interest rate as a continuous or a discrete random variable?

A. discrete
B. continuous
C. either one

Correct Answer: A

Although the interest rate is a continuous random variable, its movement is discrete because there are only 3 possible outcomes: increasing, decreasing, or staying the same.

Practice Question 20

A quoted stock price quoted in ticks of $0.01 is an example of a ______.

A. discrete random variable
B. continuous random variable
C. cumulative distribution function

Correct Answer: A

A quoted stock price quoted in ticks of $0.01 is an example of a discrete random variable.

Practice Question 21

The discrete uniform distribution ______.

A. has an infinite number of unspecified outcomes
B. is based on the Bernoulli random variable
C. has a finite number of specified outcomes

Correct Answer: C

The discrete uniform distribution is known as the simplest of all probability distributions. It is made up of a finite number of specified outcomes and each outcome is equally likely.