Random Variables in Probability Theory

Define Random Variable

A random variable is basically a mathematical postulate (rule) that allocates a numerical value to each outcome in a sample space. Random variables can either be a discrete random variable or continuous random variable. A random variable is said to be discrete when it assumes only particular values in an interval. Or else, it will be continuous. When X takes values 1, 2, 3, 4, 5, 6…, it is said to contain a discrete random variable.

Uses and Significance of Random Variable

As a function, a random variable is required to be computed, which enables probabilities to be allocated to a set of potential values. It is evident that the result is dependent upon some physical variables which are not foreseeable. Say, when we toss a fair coin, the ultimate outcome of occurring to be heads or tails will rely on the possible physical conditions. We are not able to foresee, which outcome will be noted. Although there are other probabilities like the coin could get lost or tampered, such contemplation is overlooked.

Types of Random Variables

There are mainly 2 types of random variables, as given below:

1. Continuous Random Variable

2. Discrete Random Variable

Random Variable In Probability

In probability, a real-valued function, described over the sample space of a random experiment, is known as a random variable. That is, the values of the random variable are in correspondence to the results of the random experiment. Random variables could either be continuous or discrete.

A random variable’s possible values may exhibit the possible results of an experiment, which is about to be carried out or the possible results of a preceding experiment whose existing value is not known. They may also theoretically define either the outcomes of an “objectively” random procedure (like rolling a die) or the “subjective” randomness that occurs as a consequence of insufficient knowledge of a quantity.

The domain of a random variable is a sample space, which is bespoke of an assemblage of possible outcomes from a random event. For example, when a coin is tossed, only two possible outcomes are addressed such as heads or tails.

Probability Distribution In Random Variable

A random variable and probability distribution can appear to be like:

• Conceptual recording of outcomes as well as probabilities of the outcomes.

• An experimental listing of outcomes linked with their distinguished relative frequencies.

• A subjective listing of outcomes linked with their subjective probabilities.

 The probability of a random variable X that takes the values x is described as a probability function of X which is represented by f (x) = f (X = x)

A probability distribution always fulfills two conditions that are as given:

• f(x)≥0

• ∑f(x)=1

Important Probability Distributions

Following are the important probability distributions:

• Binomial distribution

• Bernoulli’s distribution

• Exponential distribution

• Normal distribution

• Poisson distribution

Discrete Random Variable and Probability Distribution

Discrete probability distribution of a random variable X is a list of every possible value of X together with the probability that X gets hold of that value in one trial of the experiment. Each probability P(x) should be between 0 and 1 i.e.: 0≤P(x) ≤1. The sum of all the likely probabilities is 1: ∑P(x)=1.

Did You Know

• Random Variables discrete or continuous can also be transformed, meaning that the value can be reassigned to another variable.

• The transformation of random variable is actually incorporated to remap the number line from x to y

• There is also a random variable that we call a geometric random variable

Solved Examples

Example:

Calculate the mean value for the continuous random variable, when assigned the function f(x) = x, 0 ≤ x ≤ 2.

Solution:

Given function: f(x) = x, 0 ≤ x ≤ 2

Using the formula to calculate the mean value: E(X) = \[\int_{-\infty}^{\infty}\] x f(x)dx

We get,

E(X) = \[\int_{0}^{2}\] xf(x) dx

E(X) = \[\int_{0}^{2}\] x. xdx

E(X) = \[\int_{-\infty}^{\infty}\] x\[^{2}\] dx

E(X)=(x³/3) \[_{0}^{2}\]

E(X)=(2³/3)−(0³/3)

E(X)=(8/3)−(0)

E(X)=8/3

Hence, we get the mean of the continuous random variable= E(X) = 8/3.