Probability Distribution of Random Variables

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 in order 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 of the probability for each random variable value must equal one.

Types of Random Variable

As was said in the beginning, there are three types of random variables:

• Continuous Random Variable

• Discrete Random Variable

• Mixed Type

Let’s go over the different kinds of variables and give some examples of each.

Discrete Random Variable

A discrete random variable can only take on certain numbers, like 0, 1, 2, 3, 4, 5, etc. The probability mass function lists how likely each possible value in the probability distribution of a random variable is to be.

Allow a person to be picked at random and a random variable to stand in for the person’s height in an analysis. The random variable can be thought of as a function that relates to a person’s height. Concerning the random variable, the probability distribution lets you figure out the chance that the height falls within any subset of possible values, such as the chance that the height falls between 175 cm and 185 cm or between 145 cm and 180 cm. The person’s age is also a random variable. It could be between 45 and 50 years old, but it could also be less than 40 or more than 50.

Continuous Random Variable

A numeric variable is said to be continuous if it can take on the values a and b in any unit of measurement. X is said to be continuous if it can take on an infinite number of values. If X takes on any value within the interval, it is called a continuous random variable (a, b).

Random variables that don’t change over time are called continuous random variables. There are no “gaps” between the numbers that could be compared to numbers with a low chance of happening. On the other hand, these variables almost never take on a clearly defined value c, but their values are likely to be contained within small intervals.

Mixed Type

A mixed random variable is one for which the cumulative distribution function is neither discrete nor continuous everywhere. It can be seen as a mix of discrete and continuous random variables. In this case, the CDF is the weighted average of the CDFs of the individual variables.

A mixed-type random variable could be found by tossing a coin and only spinning the spinner if the coin lands on its head. If the result is a “tail,” X = 1, but if the result is a “head,” X = the value of the spinner, as in the previous case. There is a 50/50 chance that this random variable will have the value 1. In other ranges of values, the probabilities might be half as likely as in the last example.

In general, any probability distribution on the real line has a discrete part, a singular part, and a completely continuous part. See Lebesgue’s decomposition theorem for more information. § Refinement. The discrete part looks at a set that can be counted, but this set may be big (like the set of all rational numbers).

Random Variable Formula

With the formula, you can figure out the mean and variance of a set of random variables. So, we’ll talk about two important formulas here:

• Random variable mean

• Random variable variance

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

where X is a list of all the possible values and P is a list of how likely each value is.

The variance of a Random Variable: The variance of a random variable X shows how far it is from the mean value. The formula for a random variable’s variance is Var(X) = 2 = E(X₂) – [E(X)]. 2 where E(X₂) equals X₂P and E(X) equals XP

Random variables and probability distribution

Distribution of odds for a random variable

• A list of possible outcomes and how likely they are to happen

• A table with the results of an experiment and how often they happened.

• A subjective list of possible things that could happen, with probabilities that are also subjective.

A random variable X that takes the values x is said to have a probability function that looks like this

• f (x) = f (X = x).

There are always two things that a probability distribution must do

• f(x)≥0

• ∑f(x)=1

The most important probability distributions are as follows

• Binomial distribution

• Poisson distribution of probabilities

• Bernoulli’s distribution of chances

• Exponential probability distribution

• Standard deviation

Conclusion

A random variable is a number that shows how a statistical experiment turned out. A discrete random variable can only take on one of two values: a limited number or an infinite sequence of values. A continuous random variable, on the other hand, can take on any value in any interval along the real number line. For example, the number of cars sold at a certain dealership on a certain day would be a discrete random variable, while a person’s weight in kilogrammes (or pounds) would be a continuous random variable.