# Probability Distribution Function Study Notes for GATE CSE

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Probability Distribution Function Study Notes– In this article, we are going to define random variables and some related concepts such as probability mass function, expected value, variance, and median and we will look at some important types of random variables including the  Poisson, binomial, and normal. A random variable may be defined as a function on a sample space.
The values of the function can be anything but we will always take numbers. The r.v. is abbreviated as “Random Variable”.

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Example: I randomly choose a student from the class, measure and note his or her height in centimetres (cm).

Here in this example, the ‘set of students’ is sample space; the ‘height’ is a random variable which is a function to the real numbers from the set of students:  h(S) is defined as the height of student S in centimetres(cm). (NOTE: A function is a rule for associating domain set with each element to its target or range set. Here the domain set is the sample space S, the set of students in the class, and the target space is the set of real numbers.)

A random variable F will be discrete if the values taken by it are separated by gaps.
A random variable will be continuous if there are no gaps between its possible values.

## PROBABILITY MASS FUNCTION

The probability mass function of a discrete random variable F is defined as the formula, function or table which gives the value of P(F = a) for every element a in the target set of F. If F takes only some values, it is better to list them in a table; otherwise we should give a formula (if possible). The p.m.f. is abbreviated as ‘probability mass function’.

## PROBABILITY DENSITY FUNCTION

The probability density function is defined as a function of a continuous random variable, which is integrated across an interval and gives the probability that the value of the variable lies inside the same interval.

### PROBABILITY DISTRIBUTION FUNCTIONS:

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