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

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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”.

This page titled 6.3: Probability Distribution Function (PDF) for Discrete Random Variables is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Maurice A. Geraghty via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.