Probability Mass Function

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

The formula to calculate the probability mass function is:

f(x) = P(X = x)

Where,

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

X is discrete random variable

x is the 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 here:

• The PMF function follows two conditions:
   

1. The probability of each possible outcome should be non-negative.
2. The sum of all probabilities for every possible outcome 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 the 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.
  
  Because 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. On the other hand, the cumulative distribution function (CDF) is calculated by adding up the 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:

The 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 is:
 

Binomial distribution: P(X = x) = ⁿC_x × 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 Poisson distribution is:

Poisson distribution: P(X = x) = (λ^x × e^−λ) / x!

Here, the mean represented is the symbol λ.
 

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

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

We use 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.

• Doctors use 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 card out of the deck.

• It can be used to determine the probability of getting a particular number when rolling a die.

• Players use PMF to predict the scores or the number of successful shots played in the game.