PMF Full Form
What is the full form of PMF?
The Probability Mass Function, or PMF for short, is a type of frequency function or probability function that describes the distribution of a discrete random variable. In the event that X is a discrete random variable of a function, the probability mass function of X is given by -
\mathbf{P}x\left(X\right)=\mathbf{P} \left(X=x\right)
, for all x falls within the range of x.
It should be noted that the condition should satisfy the probability function:
\mathbf{P}x\left(X\right)\geq0
\sum xeRange(x)\mathbf{P}x\left ( X \right )= 1
Range(X) in this case is a countable set and can be expressed as x1, x2, x3,.... Consequently, the random variable X now has the values x1, x2, and x3, respectively.
Definition
The Probability Mass function, which accepts all arguments of any real number, is defined on all values of R. When the argument value is equal to zero, it does not belong to the value of X, and when it does, the PMF value should be positive.
A discrete probability distribution is typically defined by its probability mass function, which differs from the probability density function (PDF) in that it yields distinct results. Because of this, statistical modelling and computer programming both use the probability mass function. In other words, a probability mass function connects discrete events to the probabilities that those events will occur. The probability that is concentrated on discrete events is denoted by the word "mass."
How do a PMF and a PDF differ from one another?
The difference between PMF and PDF is
PMF | |
Numerous discrete random variables can be used in the solution. | The answer can be found in a number of continuous random variables. |
makes use of discrete random variables. | makes use of permanent random variables |
Uses for Probability Mass Functions
In statistics, the probability mass function is crucial. The probabilities for the specified discrete random variable are defined. The probability for the given random variable is equal to the variable's integration for the given random number.
It is used to figure out the discrete distribution's mean and variance.
When using discrete values, it is used in binomial and Poisson distributions to determine the probability value.
Examples of probability mass functions using Poisson and binomial distributions include the following:
PMF of Binomial Distribution
The PMF can be used in the binomial distribution for the following purposes:
To determine the number of effective sales calls
To determine how many items in the production run are defective
Counting the heads and tails when flipping a coin
Counting the number of men and women working at a company
Calculating the electoral votes for two different candidates
Think about a scenario where there are ten multiple-choice questions on a test, each with four possible answers, only one of which is correct. The probability mass function is used to calculate the likelihood of receiving both correct and incorrect answers.
PMF of Poisson Distribution
Similar to binomial, PMF is used for the distribution of poisson.
To determine the monthly requirements for a specific product
Estimate the number of customers who enter a bank each hour
Calculate the hourly access rate to a specific web server
Count the number of errors a book contains
Steps to Find the Probability Mass Function
Finding the PMF for a variable is simple. The steps you must take to determine a variable's PMF are listed below:
Step 1: Complete the first requirement of the PMF to begin answering the question.
Step 2: Add up all of the values of P (X- x). Whole numbers (0, 1, 2), variable numbers (1, 2y, 3y), and squared numbers (2y^{2} 
, 3y^{2} 
) will all be present.
Step 3: Start solving the sum using simultaneous equations.
Step 4: As soon as you begin using simultaneous equations, you will ultimately receive two answers.
Step 5: Determine which of the provided answers satisfies the following two requirements:
(i) The variable's value is always positive.
(ii) The variable's value is not equal to zero.
Step 6: The response to the query is the one that satisfies the two conditions stated above.
Frequently Asked Questions (FAQs)
The term "discrete probability distributions" refers to probability mass functions (pmf). While describing continuous probability distributions are probability density functions (pdf).
The Probability Mass function, which accepts all arguments of any real number, is defined for all values of R. When the argument value is equal to zero, it does not belong to the value of X; instead, it does belong to the value of X when the PMF value is positive.
This is how a PMF equation appears: P(X = x). Simply put, that refers to "the likelihood that X takes on some value x." It's not a very useful equation on its own; one that tells you the likelihood that a specific event will occur is more helpful.
A discrete random variable, such as a stock or ETF, as opposed to a continuous random variable, is defined by the probability density function (PDF), a statistical expression that describes a probability distribution (the likelihood of an outcome).
The kilogram (kg) is the SI unit of mass.