A uniform distribution is one for which the probability of occurrence is the same for all values of X. It is just one type of special random variable and is sometimes called a rectangular distribution. For example, if a die is thrown, the probability of obtaining any one of the six possible outcomes is 1/6. Since all outcomes are equally probable, the distribution is uniform. If a uniform distribution is divided into equally spaced intervals, there will be an equal number of members of the population in each interval.

The discrete uniform distribution is the simplest of all probability distributions. This distribution has a finite number of specified outcomes, and each outcome is equally likely. Mathematically, suppose that a discrete uniform random variable, X, has n possible outcomes: x₁, x₂, ..., x_n-1, and x_n.

• p(x₁) = p(x₂) = p(x₃) = ... = p(x_n-1) = p(x_n) = p(x). That is, the probabilities for all possible outcomes are equal.
• F(x_k) = kp(x_k). That is, the cumulative distribution function for the k^th outcome is k times of the probability of the k^th outcome.
• If there are k possible outcomes in a particular range, the probability for that range of outcomes is kp(X).

For example, the possible outcomes are the integers 1 to 8 (inclusive), and the probability that the random variable takes on any of those possible values is the same for all outcomes (i.e., it is uniform).

If a continuous random variable is equally likely to fall at any point between its maximum and minimum values, it is a continuous uniform random variable, and its probability distribution is a continuous probability distribution.

• The probability density function is: f(x) = 1/(b - a) for a ≤ x ≤ b; or 0 otherwise.
• The cumulative density function is: F(x) = 0 for x ≤ a; (x - a)/(b - a) for a ≤ x ≤ b; 1 for x ≥ b.

• The probability density function is a horizontal line with a height of 1/(b-a) over a range of values from a to b.
• The cumulative density function is a sloped line with a height of 0 to 1 over a range of values from a to b, and is a horizontal line with a height of 1 when the value of the variable equals or exceeds b.

For example, with a = 0 and b = 8, f(x) = 0.125. If this density is graphed, it will plot as a horizontal line with a value of 0.125.

To find the probability F(3) = P(X ≤ 3), find the area under the curve graphing the probability density function, between 0 and 3 on the x-axis. The middle line of the expression for the cumulative probability function is:

F(x) = 0 for x ≤ a; (x - a)/(b - a) for a < x < b; 1 for x ≥ b

For a continuous uniform random variable, the mean is given by μ = (a + b)/2 and the variance is given by σ² = (b - a)²/12.

Practice Question 1

A coating machine coats film between 120 and 210 microns with a uniform random distribution. Calculate the mean coating thickness.

A. 165 microns
B. 330 microns
C. 210 microns

Correct Answer: A

Practice Question 2

Suppose you have a discrete uniform probability function such that p(X = x) = 20% for X values of 0, 1, 2, 3, and 4. Find F(4).

A. 0%
B. 20%
C. 100%

Correct Answer: C

F(4) is the probability that the function takes on values less than or equal to 4. So, F(4) = p(0) + p(1) + p(2) + p(3) + p(4) = 0.20 + 0.20 + 0.20 + 0.20 + 0.20 = 1 = 100%.

Practice Question 3

Consider an experiment that consists of removing a card from a deck and identifying its suit. A deck has 52 cards: 13 diamonds, 13 hearts, 13 spades, and 13 clubs. Each card has an equal probability of being selected. What is the probability that a card selected randomly is a heart?

A. 0.02
B. 0.07
C. 0.25

Correct Answer: C

The probability of selecting each card from the deck randomly follows a uniform distribution. Given that there are 52 different cards, each card has a 1/52 chance of being selected. If we are only interested in selecting a card of a certain suit, then we must add the probabilities for all the cards with that suit. This probability is 13/52 = 0.25.

Practice Question 4

An experiment consists of throwing a die and recording the number of dots on the face of the die facing up. What is the probability that the number of dots is greater than 2 but less than 6?

A. 1/3
B. 1/2
C. 1/6

Correct Answer: B

Note that for a fair die, p(X) = 1/6, P(2 < X < 6) = P(X = 3) + P(X = 4) + P(X = 5) = 1/2.

Practice Question 5

A broker selected 20 different stocks that he likes equally. He identified these stocks with alphabet letters from A to T. Assume that an experiment consists of randomly selecting one of these stocks and then buying it. What is the probability that the broker buys stock V?

A. 0.05
B. 0.01
C. None of the above

Correct Answer: C

There is no stock V; the probability of selecting a nonexistent stock is 0.

Practice Question 6

We cannot count the outcomes of a ______.

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

Correct Answer: A

Practice Question 7

Suppose that the penalty for withdrawing funds early from a certain account follows a uniform distribution on the interval from (5%,12%). Find the variance of the penalty.

A. 4.08%
B. 7%
C. 8.5%

Correct Answer: A

The variance is V(X) = (12 - 5)²/12 = 4.08%.

Practice Question 8

Suppose that the penalty for withdrawing funds early from a certain account follows a uniform distribution on the interval from (5%,12%). What is the probability that the penalty is between 6% and 9%?

A. 0.143
B. 0.429
C. 0.882

Correct Answer: B

The probability is P(6 < X < 9) = F(9) - F(6) = (9-5)/7 - (6-5)/7 = 0.429.

Practice Question 9

If a continuous random variable appears to have equally likely outcomes over a range of possible values, it possesses a ______ distribution.

A. standard normal
B. uniform probability
C. Poisson

Correct Answer: B

Practice Question 10

The St. Helens Insurance Company can maintain its risk-based capital ratio, or improve upon it, if its net income is at least $125 million. If earnings can fall anywhere from -$50 million to $500 million with equal probability, what is the likelihood the company can improve its risk-based capital ratio?

A. 68.2%
B. 67.3%
C. 69.1%

Correct Answer: A

This is a continuous uniform distribution, where b = $500 million and a = -$50 million. F(x) = (x - a)/ (b - a) for a < x < b; F(x) = 0 for x <= a, and F(x) = 1 for x >= b. We are solving for 1 - F(125) = 1 - (125 - (-50))/(500 - (-50)) = 1 - 175/550 = 68.2%.

Practice Question 11

Recent history has shown that movements of more than 3% in the Dow Jones Industrial Average occur in 2% of the trading days. What is the probability that within the next 30 trading days there will be one or more days in which the average will move more than 3%?

A. 0.4545
B. 0.0836
C. 0.2876

Correct Answer: A

1 - 0.98³⁰ = 0.4545

Practice Question 12

A company is in breach of a covenant in the lending facility if the interest coverage ratio, EBITDA/Interest, falls below 3. You forecast interest of $30 million. You estimate EBITDA to be between $50 and $75 million. Assuming the outcomes for EBITDA are equally likely, what is the probability that the coverage ratio will fall below 1.8?

A. 12%
B. 15%
C. 16%

Correct Answer: C

You forecast interest of $30 million. You estimate EBITDA to be between $50 and $75 million. Assuming the outcomes for EBITDA are equally likely, the probability is 16%, and the coverage ratio will fall below 1.8.

EBITDA/Interest is a continuous uniform random variable because all outcomes are equally likely. The coverage ratio can have a range of values as follows:

EBITDA/Interest = 50/30 = 1.667 and EBITDA/Interest = 75/30 = 2.5.
Range = 2.5 - 1.667 = 0.833
The distance between 1.8 and 1.667 is 0.133.
The value of 0.133 as a percentage of 0.833 is 0.133/0.833 = 0.16 = 16%.