Probability Mass Function

Probability mass function (PMF) indicates the probability assigned to each possible value of a discrete random variable. For e.g., when we flip a coin twice, the PMF gives the probability of each possible outcome, which is heads or tails.

The formula for the Probability Mass Function:

f(x) = P(X = x)

Where: f(x) or P(X = x): the probability of a discrete random variable X 

X: discrete random variable

x: possible value that the random variable takes
 

There are unique characteristics of Probability Mass Function (PMF) that you might not know. We’ll explore a few key facts:

• The PMF function follows two conditions:
  The probability of each possible outcome should be non-negative. The sum of all probabilities for every possible should be equivalent to one.
  
   
• PMF can be used in calculating the predicted values for any random discrete variable.
  
   
• Variance of a discrete random variable can be calculated using the formula assigned for variance.
  
   
• The PMF can be visually represented on a graph. Here, the x-axis will represent the possible values of the discrete random variable and y-axis will represent their corresponding probabilities.
   

Understanding the properties of the probability mass function enables us to identify the function easily. Here are a few properties that will help:

• f(x) = P (X = x) > 0.
  
  Since the PMF cannot be negative.

• ∑x∈ S f(x) = 1       
  
  The total of all probabilities should be equivalent to 1

• P (X ∈ E) = ∑x∈ E f(x) 
  
  To calculate the probability of an event E, add up the probabilities of the values of x in E using the PMF whereas, the CDF is calculated by adding up PMF values.

The Probability Mass Function for a discrete variable X can be mathematically expressed as: f(x) = P(X = x). There are various other formulas to determine PMF for different distribution, as listed below:

Probability Mass Function in Binomial Distribution:

In binomial distribution represents the number of possible outcomes, the likelihood of success, and the likelihood of failure.

The formula we use for the binomial distribution:

P (X = x) = nCx p^x (1 - p)n-x

Here:

n: Number of outcomes

p: probability of success

(1- p): the probability of failure

Probability Mass Function in Poisson Distribution:

The Poisson distribution represents the average and the quantity of independent events that occurred during a certain period.

The formula we use for the binomial distribution:

P(X = x) = [λ^xeλ] / x!

Here, the mean represented is the symbol λ.
 

We graphically represent the probability mass function in different forms, such as tables or graphs. For example, if a coin is flipped 4 times and x is the random variable that indicates the quantity of tails. The probability for the mass function of the event:
 

Let’s now plot this in a probability mass function graph:

We use the probability mass function to understand the probability of discrete events in different real-life situations. Let’s look at some:

• PMF is used in businesses to analyze the number of customers visiting a shop within a certain period.
  
   
• The doctors use the PMF to determine the number of factors that affect a person’s disease.
  
   
• Children can use PMF to predict the probability of drawing a particular car out of the deck.
  
   
• It can be used to determine the probability of getting a particular number by rolling a die.
  
   
• Players use the PMF to predict the scores or the number of successful shots in the game they play.