Random Variable in Statistics

A random variable is a numerical representation of a statistical experiment’s outcome. A discrete random variable can only take one of two values: a finite number or an infinite succession of values; a continuous random variable can take any value in any interval on the real number line.

A sample space is the collection of alternative outcomes of a random event, and it is the domain of a random variable. When a coin is tossed, for example, there are only two possible outcomes: heads or tails.

Random Variable Types

As stated in the introduction, there are two types of random variables:

• Random Discrete Variable

• Random Continuous Variable

Let’s look at the different sorts of variables and some instances.

Random Discrete Variable

A discrete random variable has a finite number of possible values, such as 0, 1, 2, 3, 4, and so on. The probability mass function compares each of the possible values in a random variable’s probability distribution to a list of probabilities.

Allow a person to be chosen at random and a random variable to represent the person’s height in an analysis. The random variable can be logically stated as a function that ties a person’s height to themselves. In terms of the random variable, it is a probability distribution that allows the probability that the height falls into any subset of plausible values to be calculated, such as the likelihood that the height falls between 175 and 185 cm or the chance that the height falls between 145 and 180 cm. The person’s age, which could vary from 45 to 50 years old, could be less than 40 or greater than 50, which is another random variable.

Random Continuous Variable

If a numerically valued variable may take on the values a and b in any unit of measurement, it is considered continuous. If the random variable X can take on an infinite and uncountable number of values, it is considered continuous. When X takes any value inside that interval, it is considered a continuous random variable (a, b).

A continuous random variable has a cumulative distribution function that is constant throughout. There are no “gaps” between the numbers that can be compared to those that are unlikely to appear. Alternatively, these variables almost never take on an exactly defined value c, but their value is likely to fluctuate at tiny intervals.

Functions of Random Variables

If the random variable X has the values x₁, x₂,…, and the probability P (x₁), P (x₂),…, then the random variable’s expected value is:

Expectation of X, E (x) = ∑ x P (x).

A real Borel measurable function g: R→R can be applied to the outcomes of a real-valued random variable X to create a new random variable Y. Y = f, in other words (X). The following is the cumulative distribution function of Y:

F_y(y) = P(g(X)≤y)

If function g is invertible (for example, h = g^-1) and rising or decreasing, the previous relationship can be expanded to:

Fy(y) = p(g(X) ≤ y) 

1) = P(X ≤ h(y)) = Fx(h(y)), if h = g^-1 increasing,

2) = P(X ≥ h(y)) = 1 – Fx(h(y)), if h = g^-1 decreasing,

The relationship between the probability density functions can be discovered by differentiating both sides of the preceding expressions with regard to y:

f_y(y) = f_x(h(y)) |dh(y)/dy|

The formula for Random Variables

For a given set of data, the formula determines the mean and variance of random variables. As a result, two major formulas will be defined here:

• Mean of the random variable

• Variance in random variables

Mean (μ) = ∑XP, where X represents the random variable and P represents the relative probabilities.

Where X stands for all conceivable values and P stands for their relative likelihood,

The Variance of Random Variable X: The variance of random variable X reveals how far it deviates from the mean value. The formula for the variance of a random variable is Var(X) = 2 = E(X₂) – [E(X)]. E(X₂) = X₂P, and E(X) = XP.

Random Variables and Probability Distributions

A random variable’s probability distribution

• A theoretical enumeration of possible outcomes and their probabilities

• An experimental table with the observed relative frequencies of the outcomes.

• A subjective list of possible outcomes is accompanied by subjective probabilities.

The probability function for a random variable X that takes the values x is

f (x) = f (X = x).

Two requirements must always be met by a probability distribution

• f(x)≥0

• ∑f(x)=1

The most important probability distributions are as follows

• Binomial probability distribution

• Probability distribution Poisson

• The probability distribution of Bernoulli

• The probability distribution is exponential

• Descriptive statistics

Conclusion

A random variable is a numerical representation of a statistical experiment’s outcome. A discrete random variable can only take one of two values: a finite number of values or an infinite sequence of values, but a continuous random variable can take any value along the real number line. For example, a random variable representing a person’s weight in kilograms (or pounds) would be discrete. In contrast, a random variable expressing a person’s weight in kilograms (or pounds) would be continuous.