Functions of Random Variables

In order to give probabilities to a collection of probable values, a random variable must be measured as a function. Clearly, the consequences depend on a number of unforeseeable physical elements. For instance, the outcome of a fair coin flip will be determined by the physical conditions. It is impossible to determine which outcome will be remembered. Other possibilities, such as the coin shattering or vanishing, are not examined. This essay will offer a thorough understanding of Random Variables and Probability Distributions.

Types of Random Variable

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

• Discrete Random Variable

• Continuous Random Variable

Let’s review the different types of variables along with examples.

Discrete Random Variable

A discrete random variable has limited possible values, such as 0, 1, 2, 3, 4, 5, etc. The probability mass function lists probabilities related to each possible value in a random variable’s probability distribution.

Permit the random selection of a person and a random variable to represent the height of the individual in an analysis. The random variable is characterised logically as a function relating to an individual’s height. Regarding the random variable, the probability distribution enables the calculation of the probability that the height falls within any subset of plausible values, such as the probability that the height falls between 175 and 185 cm or the chance that the height falls between 145 and 180 cm. The individual’s age is also a random variable that could range from 45 to 50 years old, or be less than 40 or greater than 50.

Continuous Random Variable

A numeric variable is considered continuous if it may take on the values a and b in any unit of measurement. X is termed continuous if it can assume an unlimited and uncountable range of values. If X assumes any value inside the interval, it is regarded as a continuous random variable (a, b).

A random variable has a constant cumulative distribution function. There are no “gaps” between the numbers that can be compared to low-probability numbers. Alternately, these variables almost never adopt a precisely defined value c, but there is a significant likelihood that their values will be contained within small intervals.

Random Variable Formula

For a given data collection, the formula computes the mean and standard deviation of random variables. Therefore, we shall define two important formulas:

• Random variable mean

• Random variable variance

The mean of a random variable is defined as Mean (μ) = ∑XP where X is the random variable and P denotes the relative probabilities.

where X represents all possible values and P represents their relative probability.

Change in a Random Variable: The variance is the amount by which the random variable X deviates from its mean. Var(X) = 2 = E(X₂) – [E(X)] is the formula for the variance of a random variable. 2 where E(X₂) = X₂P and E(X) = XP.

Random Variable Distribution

The probability distribution for a random variable

• A listing of hypothetical outcomes and their probabilities

• A table listing the outcomes of an experiment and their observed relative frequencies.

• A subjective list of potential possibilities is accompanied by subjective probabilities.

f (x) = f (X = x) represents the probability function of a random variable X with the value x.

A probability distribution must always satisfy the two conditions listed below

• f(x)≥0

• ∑f(x)=1

The following are the most significant probability distributions

• Binomial distribution

• Poisson probability distribution 

• Bernoulli probability distribution

• Exponential probability distribution

• Standard deviation

Functions of Random Variables 

When working with random variables, it is common to consider their functions. Let Y be a discrete and continuous random variable, and let g be a function from R to R that we refer to as a transformation. Y might represent the height in inches of a randomly selected member of a particular population, and g could be a function that converts inches to centimetres, such as g(y) = 2.54 y. W = g(Y) is also a random variable, but its distribution (pdf), mean, variance, etc. will be different from Y. In statistics, transformations of random variables play a crucial function.

Linear Functions of Random Variables

• A function of numerous random variables is itself a random variable. We shall work exclusively with linear functions.

• Given random variables X₁, X₂,…, X_p and constants c₁, c₂,…, c_p, the linear combination of X₁, X₂,…, X_p is Y= c₁X₁ + c₂X₂ +… + c_pX_p (5-24).

Conclusion

A random variable is a quantitative representation of the outcome of a statistical experiment. A discrete random variable can assume a finite number of values or an infinite sequence of values, but a continuous random variable can assume any value in any interval along the real number line. A random variable indicating a person’s weight in kilogrammes (or pounds) would be continuous. In contrast, a random variable reflecting the number of automobiles sold at a specific dealership on a particular day would be discrete.