Note on Random Variable and Probability Distribution

Let us try to analyze and understand more on Random Variable and Probability Distribution along with their differences.

Random variable

Let S be a sample space associated with a given random experiment.

A real valued function X which assigns to each wi Î S, a unique real number.

Note:

There can be several r.v's associated with an experiment.
A random variable which can assume only a finite number of values or countably infinite values is called a discrete random variable.
e.g., Consider a random experiment of tossing three coins simultaneously. Let X denote the number of heads then X is a random variable which can take values 0, 1, 2, 3.

Continuous random variable

A random variable which can assume all possible values between certain limits is called a continuous random variable.

Discrete Probability Distribution

A discrete random variable assumes each of its values with a certain probability,
Let X be a discrete random variable which takes values x₁, x₂, x₃,…x_n where p_i = P{X = x_i}
Then
X : x₁ x₂ x₃ .. x_n
P(X): p₁ p₂ p_{3 .....} p_n
is called the probability distribution of x.

Note 1:

In the probability distribution of x

Note 2:

P{X = x} is called probability mass function.

Note 3:

Although the probability distribution of a continuous random variable cannot be presented in tabular form, it can have a formula in the form of a function represented by f(x) usually called the probability density function.

Probability distribution of a continuous random variable

Let X be continuous random variable which can assume values in the interval [a,b].
A function f(x) on [a,b] is called the probability density function if