Probability Mass Function
Probability mass function gives the probability that a discrete random variable will be exactly equal to a specific value. The probability mass function is only used for discrete random variables. For continuous random variables, the probability density function is used which is analogous to the probability mass function.
The probability mass function is also known as a frequency function. It can be represented numerically as a table, in graphical form, or analytically as a formula. In this article, we will take an indepth look at the probability mass function, its definition, formulas, and various associated examples.
What is Probability Mass Function?
Probability mass function (pmf) and cumulative distribution function (CDF) are two functions that are needed to describe the distribution of a discrete random variable. The cumulative distribution function can be defined as a function that gives the probabilities of a random variable being lesser than or equal to a specific value. The CDF of a discrete random variable up to a particular value, x, can be obtained from the pmf by summing up the probabilities associated with the variable up to x.
Probability Mass Function Definition
Probability mass function can be defined as the probability that a discrete random variable will be exactly equal to some particular value. In other words, the probability mass function assigns a particular probability to every possible value of a discrete random variable.
Probability Mass Function Example
Suppose a fair coin is tossed twice and the sample space is recorded as S = [HH, HT, TH, TT]. The probability of getting heads needs to be determined. Let X be the random variable that shows how many heads are obtained. X can take on the values 0, 1, 2. The probability that X will be equal to 1 is 0.5. Thus, it can be said that the probability mass function of X evaluated at 1 will be 0.5.
Probability Mass Function Formula
The probability mass function provides all possible values of a discrete random variable as well as the probabilities associated with it. Let X be the discrete random variable. Then the formula for the probability mass function, f(x), evaluated at x, is given as follows:
f(x) = P(X = x)
The cumulative distribution function of a discrete random variable is given by the formula F(x) = P(X ≤ x).
Probability Mass Function of Binomial Distribution
Binomial distribution is a discrete distribution that models the number of successes in n Bernoulli trials. These trials are experiments that can have only two outcomes, i.e, success (with probability p) and failure (with probability 1  p). The probability mass function of a binomial distribution is given as follows:
P(X = x) = \(\binom{n}{x}p^{x}(1p)^{nx}\)
Probability Mass Function of Poisson Distribution
Poisson distribution is another type of probability distribution. It models the probability that a given number of events will occur within an interval of time independently and at a constant mean rate. The probability mass function of Poisson distribution with parameter \(\lambda\) > 0 is as follows:
P(X = x) = \(\frac{\lambda^{x}e^{\lambda}}{x!}\)
Probability Mass Function Properties
There are three important properties of the probability mass function. With the help of these, the cumulative distribution function of a discrete random variable can be determined. The probability mass function properties are given as follows:
 P(X = x) = f(x) > 0. This implies that for every element x associated with a sample space, all probabilities must be positive.
 \(\sum_{x\epsilon S}f(x) = 1\). The sum of all probabilities associated with x values of a discrete random variable will be equal to 1.
 P(X ∈ T) = \(\sum_{x\epsilon T}f(x)\). The probability associated with an event T can be determined by adding all the probabilities of the x values in T. This property is used to find the CDF of the discrete random variable.
Probability Mass Function Representations
The probability mass function associated with a random variable can be represented with the help of a table or by using a graph. Taking the help of the coin toss example mentioned above, it can be seen that the random variable, X, represents the number of heads in the coin tosses. The sample space created is [HH, TH, HT, TT]. This shows that X can take the values 0 (no heads), 1 (1 head), and 2 (2 heads). The probabilities of each outcome can be calculated by dividing the number of favorable outcomes by the total number of outcomes. This gives us the following probabilities.
P(X = 0) = 1 / 4 = 0.25
P(X = 1) = 2 / 4 = 0.5
P(X = 2) = 1 / 4 = 0.25
These values can be presented as given below.
Probability Mass Function Table
A probability mass function table displays the various values that can be taken up by the discrete random variable as well as the associated probabilities. The pmf table of the coin toss example can be written as follows:
x  P(X = x) 

0  0.25 
1  0.5 
2  0.25 
Thus, probability mass function P(X = 0) gives the probability of X being equal to 0 as 0.25
Probability Mass Function Graph
The probability mass function graph is used to display the probabilities associated with the possible values of the random variable. A bar graph can be used to represent the probability mass function of the coin toss example as given below.
Probability Mass Function VS Probability Density Function
Probability mass function and probability density function are analogous to each other. The probability density function is used for continuous random variables because the probability that such a variable will take on an exact value is equal to 0. The differences between probability mass function and probability density function are outlined in the table given below.
Probability Mass Function  Probability Density Function 

Probability mass function denotes the probability that a discrete random variable will take on a particular value.  Probability density function gives the probability that a continuous random variable will lie between a certain specified interval. 
It is used for discrete random variables.  It is used for continuous random variables. 
It is evaluated at an exact point.  It is evaluated between a range of values. 
The formula for pmf is f(x) = P(X = x)  The formula for pdf is given as p(x) = \(\frac{\mathrm{d} F(x)}{\mathrm{d} x}\) = F'(x), where F(x) is the cumulative distribution function. 
To determine the CDF, P(X ≤ x), the probability mass function needs to be summed up to x values.  To determine the CDF, P(X ≤ x), the probability density function needs to be integrated from ∞ to x. 
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Important Notes on Probability Mass Function
 The probability that a discrete random variable, X, will take on an exact value is given by the probability mass function.
 The probability mass function formula for X at x is given as f(x) = P(X = x).
 The cumulative distribution function, P(X ≤ x), can be determined by summing up the probabilities of x values.
 The pmf can be represented in tabular form, graphically, and as a formula.
Examples on Probability Mass Function

Example 1: Given a probability mass function f(x) = bx^{3} for x = 1, 2, 3. Find the value of b.
Solution: According to the properties of probability mass function, \(\sum_{x\epsilon S}f(x) = 1\)
\(\sum_{x=1}^{3}f(x) = 1\)
b(1^{3} + 2^{3} + 3^{3})= 1
b (36) = 1
b = 1 / 36
Answer: b = 1 / 36 
Example 2: The probability mass function table for a random variable X is given as follows:
x 0 1 2 3 4 P(X = x) 0 0.1 0.2 0.3 0.4
Solution: P(X ≤ 2), can be computed by using the pmf property P(X ∈ T) = \(\sum_{x\epsilon T}f(x)\).
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= 0 + 0.1 + 0.2
= 0.3
Answer: P(X ≤ 2) = 0.3 
Example 3: Let X be a random variable with the probability mass function given as follows:
x 0 1 2 P(X = x) 2k^{2} + k  3 6k 8k^{2} + 2k + 3
Solution: We use the pmf property, \(\sum_{x\epsilon S}f(x) = 1\)
P(X = 0) + P(X = 1) + P(X = 2) = 1
2k^{2} + k  3 + 6k + 8k^{2} + 2k + 3 = 1
10k^{2} + 9k = 1
10k^{2} + 9k  1 = 0
10k^{2} + 10k  k  1 = 0
10k (k + 1)  1(k + 1) = 0
(10k  1)(k + 1) = 0
k = 0.1, 1
k = 1 cannot be considered as all probabilities lie between 0 and 1.
Thus, k = 0.1
Answer: k = 0.1
FAQs on Probability Mass Function
What is Meant by Probability Mass Function?
Probability Mass Function is a function that gives the probability that a discrete random variable will be equal to an exact value.
What is the Probability Mass Function Formula?
The formula for the probability mass function is given as f(x) = P(X = x). The pmf of a binomial distribution is \(\binom{n}{x}p^{x}(1p)^{nx}\) and Poisson distribution is \(\frac{\lambda^{x}e^{\lambda}}{x!}\).
How to Calculate the Probability Mass Function?
To calculate the probability mass function for a random variable X at x, the probability of the event occurring at X = x must be determined. After finding the probabilities for all possible values of X, a probability mass function table can be made for numerical representation.
What are the Properties of a Probability Mass Function?
There are three main properties of a probability mass function. These are given as follows:
 P(X = x) = f(x) > 0.
 \(\sum_{x\epsilon S}f(x) = 1\).
 P(X ∈ T) = \(\sum_{x\epsilon T}f(x)\).
Can the Probability Mass Function Be Greater Than 1?
The probability mass function cannot be greater than 1. However, the sum of all the values of the pmf should be equal to 1.
Can the Probability Mass Function Be Negative?
The value of the probability mass function cannot be negative. This is because the pmf represents a probability. As the probability of an event occurring can never be negative thus, the pmf also cannot be negative.
What is the Difference Between Probability Mass Function and Probability Density Function?
Probability mass function is used for discrete random variables to give the probability that the variable can take on an exact value. Probability density function is used for continuous random variables and gives the probability that the variable will lie within a specific range of values.
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