Probability Mass Function

The 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 a discrete random variable

x is the possible value that the random variable takes

Example for probability mass function

Let us consider a scenario in which we roll a fair 6-sided die. Let the random variable X be the number that we roll on the die. We know that the random variable X is discrete because a die shows only whole numbers. A Probability Mass Function (PMF) is a function that assigns a probability to each possible value of a discrete random variable, showing how likely each specific outcome is. Since the die is fair, each number on it has an equal chance of occurring. 

Therefore, the PMF is,

\(P(X=x)=\frac{1}{6}.\) Where, \(x = 1, 2, 3, 4, 5, 6.\)

This means that, 
 

• \(P(X=1)= \frac{1}{6}\)
   
• \(P(X=2)= \frac{1}{6}\)
   
• \(P(X=3)= \frac{1}{6}\)
   
• \(P(X=4)= \frac{1}{6}\)
   
• \(P(X=5)= \frac{1}{6}\)
   
• \(P(X=6)= \frac{1}{6}\)

We can verify this by confirming that each probability is between 0 and 1 and that their sum is 1.
 

• \(\text{All probabilities} = \frac{1}{6} \approx 0.166\), which is valid.
   
• \(\text{Sum of all probabilities} = \frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} = 1\)
   

Therefore, we can confirm that this probability mass function is valid.

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 for the binomial probability mass function is:
 

Binomial distribution: \( P(X = x) = \binom{n}{x} \, p^{x} \, (1 - p)^{\,n - x} \)

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: \(P(X = x) = \frac{(λ^x \times e^{−λ})}{x!}\)

Here, the mean represented is the symbol λ.

A table of the Probability Mass Function (PMF) lists all possible values of a discrete random variable and their associated probabilities, thereby showing students how probability is distributed across outcomes. One of the advantages of having a PMF table for students is that it allows them to visualize probability more readily, see how probabilities are assigned, and compare the likelihoods of different outcomes.

The PMF table for the number of candies a kid may pick from a jar containing two chocolates, three toffees, and five jelly candies is given as;
 

Let X be the type of candy picked.

Total candies = 10

Let us check their validity:

All probabilities are greater than or equal to 0, and the sum of all the probabilities is 1. This table helps us understand how likely each candy type is.

A Probability Mass Function (PMF) graph is a bar chart that displays the possible values of a discrete random variable along the x-axis and their corresponding probabilities on the y-axis. Here, the height of each bar represents how likely that outcome is to occur.

Let us use the same data from the above section and create a graph, as we did with the table. Therefore, the PMF graph for the number of candies a kid may pick from a jar containing two chocolates, three toffees, and five jelly candies is given as;

The probability density function (PDF) is a statistical technique for measuring the likelihood of different results across a continuous range or interval, where, rather than more discrete points on the scale, “a student’s t-distribution” is used extensively by analysts in finance to analyze distributions of return from investment and associated risk. It allows financial analysts to model and analyze risks by assuming continuous distributions of returns despite actual discrete outcomes. 

The sequential plot of the S&P 500 index over 3 years is shown in the image below. The chart showed a right-skewed bell curve, suggesting greater upside over three years.

The difference between the probability mass function and the probability density function is shown in the table below:
 

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.

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:
  
  The probability of each possible outcome should be non-negative.
  
  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.

The probability mass function can be confusing for many learners. Here are some practical, easy tips and tricks to help learners master the concept of the probability mass function.
 

• Students grasp new concepts very quickly when they can apply the idea to many familiar situations. Teachers can help students see the randomness before learning the math. Teach them concepts using a rolling die, a color wheel, picking candies from a jar, etc.
   
• Before teaching them formulas, parents can teach the concepts using many objects, such as coins, marbles, cards, and chocolates. Let the students count by touching the objects.
   
• Use bar graphs to visualize the probability mass function. It is easier for learners to understand from pictures than from equations.
   
• Compare and identify the difference between a common probability function and standard bar charts. Ask your students to identify the simple connection between them. Tell them that a bar chart shows counts, while a probability mass function graph shows probabilities.
   
• Parents can connect the probability mass function to everyday language, which would help students understand it by talking about it. Describe PMF in simple terms, like “this outcome is most likely, this one hardly happens, these two outcomes are equally likely.”
   
• Teachers can use real data collected directly from students. Ask them to collect data, such as counting the number of siblings in the class or recording shoe sizes, to create a PMF from the class data.

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.