Every random variable is associated with a probability distribution that describes the variable completely. A probability function is one way to view a probability distribution. It specifies the probability that the random variable takes on a specific value; P(X = x) is the probability that a random variable X takes on the value x.

A probability function has two key properties:

• 0 ≤ P(X=x) ≤ 1, because probability is a number between 0 and 1.

• ΣP(X=x) = 1. The sum of the probabilities P(X=x) over all values of X equals 1. If there is an exhaustive list of the distinct possible outcomes of a random variable and the probabilities of each are added up, the probabilities must sum to 1.

The following examples will utilize these two properties in order to examine whether they are probability functions.

Example 1

p(x) = x/6 for X = 1, 2, 3, and p(x) = 0 otherwise

• Substituting into p(x): p(1) = 1/6, p(2) = 2/6 and p(3) = 3/6
  Note that it is not necessary to substitute in any other values, as p(x) is only non-zero for X values 1, 2 and 3.
  In all 3 cases, p(x) lies between 0 and 1, as 1/6, 2/6 and 3/6 are all values in the range 0 to 1 inclusive.
  So, the first property is satisfied.
• Summing the probabilities gives 1/6 + 2/6 + 3/6 = 1, showing the second property is also satisfied.

Example 2

p(x) = (2x - 3)/16 for X = 1, 2, 3, 4 and p(x) = 0 otherwise

Substituting into p(x): p(1) = -1/16

STOP HERE!

It is impossible for any probability to be negative, so it's not necessary to continue. Property 1 is violated, so it can be said straightaway that p(x) is not a probability function.

Note that individual probabilities in a continuous case cannot occur, so P(X = 5), say, is 0 if X is continuous.

In a continuous case, only a range of values can be considered (that is, 0 < X < 10), whereas in a discrete case, individual values have positive probabilities associated with them.

For a discrete random variable, the shorthand notation is p(x) = P(X = x). For continuous random variables, the probability function is denoted f(x) and called probability density function (pdf), or just the density. This function is effectively the continuous analogue of the discrete probability function p(x).

• The probability density function, which has the symbol f(x), does not give probabilities, despite its name. Instead, it is the area between the graph and the horizontal axis that gives probabilities. Because of this, the height of f(x) is not restricted to the range 0 to 1, and the graph, which in itself is not a probability, is unrestricted as far as its height is concerned.

• From this information, it follows that the area under the entire graph (i.e., between the graph and the x-axis) must equal 1, because this area encapsulates all the probability contained in the random variable. Recall that for discrete distributions, the probabilities add up to 1.

• Because continuous random variables are concerned with a range of values, individual values have no probabilities, because there is no area associated with individual values. Rather, probabilities are calculated over a range of values. Another way of saying this is that p(x) = 0 for every individual X.

• If a discrete random variable has many possible outcomes, then it can be treated as a continuous random variable for conciseness, and ranges of values can be considered in determining probabilities.

Practice Question 1

For the density curve displayed below, which of the following is (are) true?

I. The mean is larger than the median.
II. The proportion of outcomes between 0.2 and 0.5 is equal to 0.3.
III. The proportion of outcomes exceeding 1.5 is equal to 0.25.

Correct Answer: III only

The proportion is the area under the density curve between 1.5 and 2. The length is 0.5, but the height is only 0.5, so the overall area is 0.5 x 0.5 = 0.25.
Because the density curve is symmetrical, the mean and median must coincide.

Practice Question 2

Let the random variable X be a random number with the uniform density curve given below.

P(0.7 < X < 1.1) has value ______.

A. 0.30
B. 0.40
C. 0.70

Correct Answer: A

Because the density ends at 1, there is no probability associated with X values greater than 1. So P(0.7 < X < 1.1) is the same as P(0.7 < X < 1.0). If you shade this region in the density, the result is a rectangle with a length 1.0 - 0.7 = 0.3 and a height of 1.0, so the area is 0.3 x 1.0 = 0.3.

Practice Question 3

The probability density function for a continuous uniform distribution is expressed as ______.

A. f(x) = 1/(b-a), for a ≤ x ≤ b, and f(x) = 1 otherwise
B. f(x) = 1/a, for x ≥ a, and f(x) = 0 otherwise
C. f(x) = 1/(b-a), for a ≤ x ≤ b, and f(x) = 0 otherwise

Correct Answer: C

By definition, f(x) = 1/(b-a) for a ≤ x ≤ b, and f(x) = 0 otherwise.

Practice Question 4

The random variable X has the following distribution:
f(x) = [c(6 - x)]/10, x = 0,1,2,3

What value of c makes this a legitimate probability distribution?

A. 3/10
B. 5/9
C. 18/10

Correct Answer: B

For this to be a legitimate probability distribution, Σ p(x) = 1, so c = 5/9.

Practice Question 5

The random variable, x, is uniformly distributed over the range of 0 to +10. The probability of observing a value of x=4 is closest to ______.

A. 0%
B. 10%
C. 40%

Correct Answer: A

The probability of observing a specific value for a continuous random variable is zero.

Practice Question 6

Which of the following is (are) NOT (a) key property (properties) of a probability function?

I. Each individual probability must lie between 0 and 1 inclusive.
II. The sum of all probabilities over the entire range of the random variable must equal 1.

A. I only.
B. II only.
C. None of the above.

Correct Answer: C

Practice Question 7

That which specifies the probability that the random variable takes on a specific value is known as the ______.

A. probability function
B. probability density function
C. cumulative distribution function

Correct Answer: A

The probability function specifies the probability that the random variable takes on a specific value.

Practice Question 8

A certain function has 2 key properties:

I. 0 <= p(x) <= 1.
II. the sum of the probabilities p(x) over all values of X equals 1.

This function is known as the ______.

A. probability function
B. probability density function
C. discrete uniform distribution

Correct Answer: A

This function is known as the probability function.