Table of Contents
It’s effortless to find the PMF for a variable. Given below are the steps that you need to follow to find the PMF of a variable:
Step 1: Calculate the mean of the variable.
Step 2: Calculate the standard deviation of the variable.
Step 3: Divide the standard deviation by the square root of the number of observations.
Step 4: Multiply the result by the probability of an event that is less than or equal to the value of the variable.
Step 5: Add the result to 1.
Probability Mass Function Definition
A probability mass function (pmf) is a function that assigns a probability to each possible outcome of a random event. The function maps outcomes to real numbers in the range 0 to 1, with the sum of the probabilities for all outcomes equaling 1.
For example, consider the roll of a fair six-sided die. The outcomes of the roll are 1, 2, 3, 4, 5, and 6, each of which has a probability of 1/6. The pmf for this random event can be represented by the table below.
Outcome Probability 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6
The pmf can also be represented by a graph, as shown below.
Probability Mass Function
The probability mass function (PMF) is a function that assigns a probability to each possible outcome of a random event. The PMF can be used to calculate the probability of a particular outcome occurring.
The PMF for a random event can be represented by a histogram. The height of each bar in the histogram corresponds to the PMF for that outcome.
Finding The Probability Mass Function
The probability mass function (pmf) of a discrete random variable tells us how likely it is for the variable to take on a particular value. In other words, it tells us the probability of each value occurring.
There are a few steps we need to take in order to find the pmf of a discrete random variable.
1. First, we need to list all of the possible values the variable can take on.
2. Next, we need to find the probability of each value occurring.
3. Finally, we need to add up all of the probabilities to find the pmf.
Let’s look at an example.
Suppose we are interested in the probability of a die landing on a particular number.
There are six possible values the die can land on, so we list them below:
1 2 3 4 5 6
We can find the probability of each value occurring by dividing the number of times the value appears on the die by the total number of rolls of the die.
1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6
We can add up all of the probabilities to find the pmf.
1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 1/6 2/36 3/36 4/36 5/36 6
Probability Mass Function Applications
A probability mass function (PMF) is a function that assigns a probability to each possible outcome of an event. In other words, it is a way of computing the probability of an event.
There are many different applications for probability mass functions. Some of these applications include:
1. Comparing the probabilities of different events
2. Determining the odds of an event
3. Calculating the expected value of a random variable
4. Assessing the fairness of a game or gambling situation
Probability Mass Function Solved Example
The probability mass function of a random variable X is given by
f(x) = {0.1, 0.2, 0.3, 0.4, 0.5}
x
{
0.1
,
0.2
,
0.3
,
0.4
,
0.5
}
What is the probability that X is less than or equal to 0.2?
The probability that X is less than or equal to 0.2 is 0.2.
