Types of Random Variable

A random variable’s probability distribution represents the distribution of probabilities across its values. The probability distribution of a discrete random variable, x, is characterised by the probability mass function, indicated by f. (x). This function reports the chance that each random variable’s value is true. The probability function for a discrete random variable must meet two conditions: (1) f(x) must be nonnegative for each random variable value, and (2) the total of the probabilities for each random variable value must equal one.

Changes in Random Variable

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

• Continuous Random Variable

• Discrete Random Variable

• Mixed Type

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).

Continuous random variable: A random variable having a constant cumulative distribution function. There are no “gaps” between the numbers that can be compared to low-probability numbers. Alternately, these variables rarely adopt a precisely defined value c, but there is a significant likelihood that their values will be contained within small intervals.

Mixed Type

A random variable whose cumulative distribution function is neither discrete nor everywhere continuous is a mixed random variable. It can be realised as a mixture of discrete and continuous random variables; in this case, the CDF is the weighted average of the CDFs of the component variables.

A random variable of mixed type would be determined via an experiment in which a coin is tossed and the spinner is only spun if the result is headed. X = 1 if the result is a tail; otherwise, X equals the value of the spinner, as in the preceding case. The likelihood that this random variable will have the value 1 is one-half. Other value ranges might have probabilities that are half as likely as the preceding example.

In general, any probability distribution on the real line consists of a discrete component, a singular component, and a continuous component; see Lebesgue’s decomposition theorem. § Refinement. The discrete component focuses on a countable set, yet this set may be dense (like the set of all rational numbers).

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.

The variance of 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

Probability and 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

• The Poisson probability distribution

• The probability distribution of Bernoulli

• Exponential probability distribution

• Normal deviation

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. On the other hand, a random variable that shows how many cars were sold at a certain dealership on a certain day would be discrete.