PMF or probability mass function is a simple concept in mathematics. It is a part of statistics. When you are learning about pmf you will find it very interesting and informative. It's an informative and useful concept. Probability Mass Function is otherwise referred to as Probability Function or frequency function. PMF characterizes the distribution of a discrete variable which is unplanned or random. Example of a discrete random variable:
Let Y be the random variable of a function, and this is its probability mass function:
Py (y) = P (Y-y), for all y belongs to the range of Y.
Here are two conditions on which the probability function should fall upon:
• P y (y) ≥ 0
• ∑ yϵRange (y) P y (y) = 1
Probability Mass Function Definition
The definition of Probability Mass Function is that it’s all the values of R, where it takes into argument any real number. There are two times when the cost doesn't belong to Y. First, when the case is equal to zero. The second time is when the value is negative, the value of the probability function is always positive.
Another name of PMF is the Probability Discrete Function (PDF). It's given because when you are drawing the variable, it produces distinct outcomes or results. Two places where the discrete probability function is used is computer programming and statistical modelling.
Probability Mass Function
The simple meaning of Probability Mass Function is the function relating to the probability of those events taking place or occurring. The word ''mass'' is used to denote the expectations of discrete events.
Finding The Probability Mass Function
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: Start solving the question by fulfilling the first condition of the PMF. ( mentioned above)
Step 2: Take all the values of P ( X- x) and add it up. There will be a whole number ( 0, 1, 2), numbers with variables ( 1y, 2y 3y) and numbers which are squared ( 2 y2, 3 y2 ).
Step 3: Start using simultaneous equations to solve the sum.
Step 4: As you start using simultaneous equations, you will get two answers in the end.
Step 5: You need to check which of the answers fulfils these two conditions:
(i) The value of the variable is never negative.
(ii) The amount of the variable does not equal zero.
Step 6: The answer to the question is the one that follows both the conditions which are mentioned above.
Probability Mass Function Applications
There are many places where the probability mass function is used and applied. Here are some of the places where there's an application of PMF:
• One of the sections where PMF is used is statistics. It plays a vital and essential role in the study of statistics. Probability Function shows the various probabilities of the discrete variable data.
• PMF combines the variable for the random number that is identical or equal to the expectation for the random variable.
• Many people use PMF to calculate two main concepts in statistics- mean and discrete distribution.
• Another place where PMF is binomial and Poisson distribution is to find the value of the variables which are distinct and random.
There are mainly two differences between PDF and PMF. Here are the two dissimilarities between them:
Distinction Between PDF and PMF:
Probability Mass Function Solved Example
Here is a probability mass function example which will help you get a better understanding of the concept of how to find probability mass function.
Solved Example 1:
Let X be a random variable, and P (X=x) is the PMF given below;
1. Find the value of k.
2. Find the pmf probability of
(i) P (X ≤ 6 )
Solution:
Given: ∑P(xi)=1
1. So,
0+k+2k+2k+3k+ k2+ 2 k2+k = 1
9k 10 k2 = 1
9k 10 k2 - 1 = 0
10k 2 + 10k - k - 1 = 0
10 k ( k + 1) - 1 (k - 1) = 0
(10k - 1 ) ( k +1 ) = 0
Therefore,
10k - 1 = 0 and k + 1 = 0
Hence,
k= 1/ 10, and k = -1
K= -1 is not the desired answer because the pmf probability value lies between 0 and 1 ( it’s one of the conditions which is mentioned above)
Conclusion: The value of k is 1/10.
2. (i) P (X ≤ 6 )
So,
P (X ≤ 6 ) = 1 - P ( x > 6)
= 1 - 7 k2 + k
= 1 - ( 7 ( 1/10) 2 + (1 / 10))
= 1 - ( 7/ 100 + 1 / 10)
= 1 - ( 17/ 100)
= ( 100 - 17 )/ 100
= 83/ 100
Conclusion : P (X ≤ 6 ) = 83/ 100.
FAQs on Probability Mass Function
1. What is a Probability Mass Function and how is it different from a Probability Density Function?
A Probability Mass Function (PMF) describes the probability that a discrete random variable takes a specific value. The sum of probabilities assigned to all possible values equals 1. In contrast, a Probability Density Function (PDF) applies to continuous random variables and uses integration instead of summation. PMF is used for discrete data, while PDF is used for continuous data.
2. What are the essential properties that any Probability Mass Function must satisfy?
- The probability of each possible value must be between 0 and 1 (inclusive).
- The sum of all assigned probabilities over the complete set of possible values is always 1.
- No probability value can be negative or greater than 1.
3. Why does the Probability Mass Function only apply to discrete random variables?
The Probability Mass Function is defined only for discrete random variables because it assigns probabilities to distinct, separate values. Continuous random variables, on the other hand, require a probability density function as their outcome range is infinite or uncountable.
4. How is the value of a constant 'k' determined in a given PMF question?
To determine the value of k in a PMF, sum all the probabilities (using k as a variable) and set the sum equal to 1. Solve the resulting equation for k, ensuring that the solution makes all probabilities valid (i.e., non-negative and less than or equal to 1).
5. In CBSE exams, what types of problems are often asked from the Probability Mass Function chapter?
- Calculation of unknown constants in a PMF for given random variables
- Problems requiring students to verify the PMF properties
- Finding the probability of events like P(X ≤ a), P(X = a), or P(X > a)
- Application of PMF in real-life scenarios, usually statistics
6. How does understanding PMF help in solving binomial and Poisson distribution problems?
Both binomial and Poisson distributions are special cases of discrete probability distributions where PMFs are used to model the probability of a number of events occurring in a fixed interval. Understanding PMF makes it easier to directly apply the relevant distribution formulas and interpret statistical outcomes.
7. Can a PMF assign a probability value of zero to a possible outcome?
Yes, a PMF can assign a probability of zero to an outcome, meaning that the outcome is possible in theory but does not occur in practice for the given random variable.
8. What steps should be followed to verify whether a given function constitutes a valid Probability Mass Function?
- Ensure the set of all possible values of the random variable is discrete and clearly defined.
- Check every probability is between 0 and 1, inclusive.
- Sum the probabilities for all values and confirm the total is exactly 1.
9. Why is it important that the sum of all probabilities in a PMF equals one?
This condition guarantees that one of the possible outcomes must occur when the experiment is performed, ensuring logical consistency with foundational principles of probability theory.
10. What common mistakes should students avoid when working with Probability Mass Functions in board exams?
- Forgetting to include all possible values of the random variable
- Assigning negative probabilities or probabilities greater than 1
- Not verifying that the total probability sums to 1
- Confusing PMF with PDF when the variable is not discrete
