Random Variable

A real-valued function, defined over the sample space of a random experiment, is called a random variable. That is, the values of the random variable correspond to the outcomes of the random experiment. Random variables may be either discrete or continuous. In this article, let’s discuss the different types of random variables.

Random Variable Definition

A random variable is a rule that assigns a numerical value to each outcome in a sample space. Random variables may be either discrete or continuous. A random variable is said to be discrete if it assumes only specified values in an interval. Otherwise, it is continuous. When X takes values 1, 2, 3, …, it is said to have a discrete random variable.

Variate

A variate can be defined as a generalization of the random variable. It has the same properties as that of the random variables without stressing to any particular type of probabilistic experiment. It always obeys a particular probabilistic law.

  • A variate is called discrete variate when that variate is not capable of assuming all the values in the provided range.
  • If the variate is able to assume all the numerical values provided in the whole range, then it is called continuous variate.

Continuous Random Variable

A numerically valued variable is said to be continuous if, in any unit of measurement, whenever it can take on the values a and b. If the random variable X can assume an infinite and uncountable set of values, it is said to be a continuous random variable. When X takes any value in a given interval (a, b), it is said to be a continuous random variable in that interval.

Discrete Random Variable

A discrete random variable can take only a finite number of distinct values such as 0, 1, 2, 3, 4, … and so on. The probability distribution of a random variable has a list of probabilities compared with each of its possible values known as probability mass function.

Functions of Random Variable

Let the random variable X assume the values x1, x2, …with corresponding probability P (x1), P (x2),… then the expected value of the random variable is given by

Expectation of X, E (x) = ∑ x P (x).

Random Variable and Probability Distribution

The probability distribution of a random variable can be

  • Theoretical listing of outcomes and probabilities of the outcomes.
  • An experimental listing of outcomes associated with their observed relative frequencies.
  • A subjective listing of outcomes associated with their subjective probabilities.

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

A probability distribution always satisfies two conditions:

  • f(x)≥0
  • ∑f(x)=1

The important probability distributions are:

  • Binomial distribution
  • Poisson distribution
  • Bernoulli’s distribution
  • Exponential distribution
  • Normal distribution

Transformation of Random Variables

The transformation of a random variable means to reassign the value to another variable. The transformation is actually inserted to remap the number line from x to y, then the transformation function is y = g(x).

Transformation of X or Expected Value of X for a Continuous Variable

Let the random variable X assume the values x1, x2, x3, ..… with corresponding probability P (x1), P (x2), P (x3),……….. then the expected value of the random variable is given by

Expectation of X, E (x) = ∫ x P (x)

Random Variable Example

Question:

Find the mean value for the continuous random variable, f(x) = x, 0 ≤ x ≤ 2.

Solution:

Given: f(x) = x, 0 ≤ x ≤ 2.

The formula to find the mean value is \(E(X)=\int_{-\infty }^{\infty }x f(x)dx\) \(E(X)=\int_{0}^{2 }x f(x)dx\) \(E(X)\int_{0}^{2 }x.xdx\) \(E(X)\int_{-\infty }^{\infty }x^{2}dx\) \(E(X)=\left (\frac{x^{3}}{3} \right )_{0}^{2}\) \(E(X)=\left (\frac{2^{3}}{3} \right )- \left (\frac{0^{3}}{3} \right )\) \(E(X)=\left (\frac{8}{3} \right )- \left (0\right )\) \(E(X)=\frac{8}{3}\)

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

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