6.3: Probability Distribution Function (PDF) for Discrete Random Variables

All random variables have the value assigned in accordance with a probability model. For discrete variables, this assigning of probabilities to each possible value of the random variable is called a probability distribution function, or PDF for short.  

This probability distribution function is written as \(P(X=x)\) or \(P(x)\) for short. This PDF can be read as “The probability the random variable \(X\) equals the value \(x\).”

Additionally, probability statements can be written as inequalities.

\(P(X < x)\) means the probability the value of the random variable is less than \(x\).

\(P(X \leq x)\) means the probability the value of the random variable is at most \(x\).

\(P(X > x)\) means the probability the value of the random variable is more than \(x\).

\(P(X \geq x)\) means the probability the value of the random variable is at least \(x\).

Like any function in Mathematics, a probability distribution function can be defined by a description, a table, a graph or a formula. The general method of assigning probabilities to values follows this procedure.

Procedure for creating a discrete probability distribution function

1. Define the random Variable \(X\)
2. List out all possible values
3. Assign probabilities to each value. You can use counting methods or relative frequencies.
4. This assignment must follow these two rules: \(P(x) \geq 0\) and \(\sum P(x)=1\)

Example: Flip two coins

Two coins are flipped and the number of heads are counted.  

\(X\) = the number of heads when two coins are flipped

Possible Values = {0, 1, 2}

Here are 5 possible probability distribution functions:

A	B	C	D	E
\(x\) \(P(x)\) 0 1/3 1 1/3 2 1/3	\(x\) \(P(x)\) 0 0.25 1 0.50 2 0.25	\(x\) \(P(x)\) 0 0 1 0 2 1	1\(x\) \(P(x)\) 0 0.3 1 0.3 2 0.3	\(x\) \(P(x)\) 0 0.6 1 -0.1 2 0.5

Models A, B and C are valid because each probability assignment is non‐negative and all probabilities total to 1.

Model B is the correct model for flipping fair coins as there are two ways to get one head.

Model C (a coin that only comes up head) is valid since zero probability is allowed.

Model D is invalid since the probabilities do not total to 1.

Model E is invalid because negative probabilities are not allowed.

Example: Multiple choice test

Students are given a multiple choice exam with 4 questions.  

The random variable X = the number answers correct. Possible values = {0, 1, 2, 3, 4}

From past data, 10% of students get zero correct answers, 10% get exactly one correct answer, 20% get two correct, and 40% get three correct. Since the probabilities must add to 1, it can be determined that 20% of students got all correct, and the PDF can be finished.

\(x\)	\(P(x)\)
0	0.1
1	0.1
2	0.2
3	0..4
4	0.2

Solution

We can use the table to answer any type of probability question:

The probability of exactly 2 questions correct:  \(P(X =2) = P(2) = 0.2\)

The probability of fewer than 2 questions correct:  \(P(X < 2) = P(0) + P(1) = 0.1 + 0.1 = 0.2\)

The probability of more than 2 questions correct:  \(P(X > 2) = P(3) + P(4) = 0.4 + 0.2 = 0.6\)

The probability of at least 2 questions correct:  \(P(X \geq 2) = P(2) + P(3) + P(4) = 0.2 + 0.4 + 0.2 = 0.8\)

The probability of at most 2 questions correct:  \(P(X \leq 2) = P(0) + P(1) + P(2) = 0.1 + 0.1 + 0.2 = 0.4\)

The probability at least 1 question correct:  \(P(X >1) = 1 – P(0) = 1 – 0.1 = 0.9\)

The last example was done using the Rule of Complement. The complement of “at least one correct answer” is “zero correct answers”.