What is Random Variable Meaning

A Random Variable Probability is a mathematical concept that assigns numerical values to outcomes of a sample space. They can describe the outcomes of objective randomness (like tossing a coin) or subjective randomness(results of a cricket game).

There are two types of Random Variables- Discrete and Continuous.

A random variable is considered a discrete random variable when it takes specific, or distinct values within an interval. Conversely, if it takes a continuous range of values, then it is classified as a continuous random variable.

Random variables are generally represented by capital letters like X and Y. This is explained by the example below:

Example

If two unbiased coins are tossed then find the random variable associated with that event.

Solution:

Suppose Two (unbiased) coins are tossed

X = number of heads. [X is a random variable or function]

Here, the sample space S = {HH, HT, TH, TT}

Random Variable Definition

We define a random variable as a function that maps from the sample space of an experiment to the real numbers. Mathematically, Random Variable is expressed as,

X: S →R

where,

• X is Random Variable (It is usually denoted using capital letter)
• S is Sample Space
• R is Set of Real Numbers

Suppose a random variable X takes m different values i.e. sample space

X = {x₁, x₂, x₃………x_m} with probabilities

P(X = x_i) = p_i

where 1 ≤ i ≤ m

The probabilities must satisfy the following conditions :

• 0 ≤ p_i ≤ 1; where 1 ≤ i ≤ m
• p₁ + p₂ + p₃ + ……. + p_m = 1 Or we can say 0 ≤ p_i ≤ 1 and ∑p_i = 1

Hence possible values for random variable X are 0, 1, 2.

X = {0, 1, 2} where m = 3

• P(X = 0) = (Probability that number of heads is 0) = P(TT) = 1/2×1/2 = 1/4
• P(X = 1) = (Probability that number of heads is 1) = P(HT | TH) = 1/2×1/2 + 1/2×1/2 = 1/2
• P(X = 2) = (Probability that number of heads is 2) = P(HH) = 1/2×1/2 = 1/4

Here, you can observe that, (0 ≤ p₁, p₂, p₃ ≤ 1/2)

p₁ + p₂ + p₃ = 1/4 + 2/4 + 1/4 = 1

For example,

Suppose a dice is thrown (X = outcome of the dice). Here, the sample space S = {1, 2, 3, 4, 5, 6}. The output of the function will be:

• P(X=1) = 1/6
• P(X=2) = 1/6
• P(X=3) = 1/6
• P(X=4) = 1/6
• P(X=5) = 1/6
• P(X=6) = 1/6

A random variable in statistics is a function that assigns a real value to an outcome in the sample space of a random experiment. For example: if you roll a die, you can assign a number to each possible outcome.

Random variables can have specific values or any value in a range.

There are two basic types of random variables,

• Discrete Random Variables
• Continuous Random Variables

In this article, we will learn about random variable statistics, their types, random variable example, and others in detail.