Random Variables and Probability Distributions

The probability distribution of a random variable describes how probabilities are spread over the random variable’s values. The probability distribution of a discrete random variable, x, is characterised by a probability mass function, represented as f(x). This function gives back the chance that each random variable value is true. Two conditions must be met to build the probability function for a discrete random variable: (1) f(x) must be nonnegative for each value of the random variable, and (2) the total probability for each random variable value must equal one.

Variate

A variate is simply a generalisation of a random variable. It has the same qualities as random variables without being restricted to any certain sort of probabilistic experiment. It always follows a certain probability rule.

• A variate is referred to as a discrete variate if it is incapable of adopting all of the values in the given range.

• The term “continuous variate” refers to a variate that can assume all of the numerical values in a given range.

Types of Random Variables

There are two types of random variables:

• Discrete Random Variable

• Continuous Random Variable

Let’s take a closer look at various types of variables, along with examples.

Discrete Random Variable

A discrete random variable can only have a limited number of different values, such as 0, 1, 2, 3, 4, etc. The probability mass function is a list of probabilities compared to each of the possible values in a random variable’s probability distribution.

Allow a person to be chosen at random in the analysis, and a random variable represents the person’s height. The random variable is logically described as a function that connects the individual to their height. In terms of the random variable, it is a probability distribution that allows the calculation of the probability that the height falls into any subset of likely values, such as the likelihood that the height falls between 175 and 185 cm, or the possibility that the height falls between 145 and 180 cm. Another random variable is the person’s age, which could range from 45 to 50 years old, and could be less than 40 or greater than 50.

Continuous Random Variable

A numerically valued variable is considered to be continuous if it can take on the values a and b in any unit of measurement. The random variable X is considered to be continuous if it can take on an unlimited and uncountable set of values. X is considered to be a continuous random variable in that interval when it takes any value in the interval (a, b).

A continuous random variable is one with a constant cumulative distribution function throughout. There are no “gaps” between the numbers that may be compared to numbers with a low likelihood of appearing. Alternatively, these variables nearly never assume an exactly defined value c, but there is a positive likelihood that their value will rest in small intervals.

Random Variable Formula

The formula calculates the mean and variance of random variables for a given collection of data. As a result, we’ll define two major formulas here:

• 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 represents the relative probabilities.

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

The random variable variance indicates how far random variable X deviates from the mean value. Var(X) = 2 = E(X2) – [E(X)] is the formula for the variance of a random variable. 2 where E(X2) equals X2P and E(X) equals XP

Probability Distribution and Random Variable

The probability distribution of a random variable:

• Theoretical listing of outcomes and probabilities of the outcomes 

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

• A subjective list of possible possibilities with subjective probabilities.

A probability function of a random variable X that takes the values x is given by

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

A probability distribution must always meet two requirements:

• f(x)≥0

• ∑f(x)=1

The following are the most important probability distributions:

• Binomial distribution

• Poisson probability distribution

• Bernoulli’s probability distribution

• Exponential probability distribution

• Standard deviation

Conclusion

A random variable is a quantitative representation of the outcome of a statistical experiment. A discrete random variable can only assume one of two values: a limited number 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 reflecting the number of automobiles sold at a certain dealership on a given day, for instance, would be discrete, whereas a random variable expressing a person’s weight in kilogrammes (or pounds) would be continuous.