Probability Distribution

The probable result of a random experiment is called an outcome. And a set of outcomes is called a sample point. For additional educational resources For additional educational resources, visit the U.S. Department of Education website.

With the help of these experiments or events, we can always create a probability pattern table in terms of variables and probabilities. For additional educational resources

Probability distribution of random variables

A random variable has a probability distribution that specifies the probability of its unknown values. Random variables can be discrete (not constant) or continuous, or both. That means it takes any of a designated finite or countable list of values, provided with a probability mass function feature of the random variable’s probability distribution, or can take any numerical value in an interval or set of intervals.

Two random variables with equal probability distribution can yet be different in their relationships with other random variables or whether they are independent of these.

Detecting a random variable means that the outcomes of randomly selecting values based on the variable probability distribution function are called random variables.

Types of the probability distribution

There are two types of probability distributions that are used for different purposes and different types of data production processes.

1. Normal or cumulative probability distribution
2. Binomial or discrete probability distribution

Cumulative probability distribution:

Cumulative probability distributions are also known as continuous probability distributions. In this distribution, a set of possible outcomes can take values in a continuous range. For example, a set of real numbers, is a continuous or normal distribution, as it gives all the possible outcomes of real numbers.

The formula for the normal distribution is:

\(\color{blue}{P\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}\left(\frac{x-\mu }{\sigma }\right)^2}}\)

Where,

• \(\mu =\) Mean value
• \(σ =\) Standard distribution of probability
• \(x =\) Normal random variable

If mean \((μ) = 0\) and standard deviation \((σ) = 1\), then this distribution is known to be a normal distribution.

Discrete probability distribution:

A distribution is called a discrete probability distribution, in which a set of outcomes is discrete in nature. For example, if a dice is rolled, then all the possible outcomes are discrete and give a mass of outcomes. It is also known as the probability mass function.

Therefore, the outcomes of a binomial distribution include n repeated experiments, and the outcome may or may not occur. The formula for the binomial distribution is:

\(\color{blue}{P\left(x\right)=\frac{n!}{r!\left(n-r\right)!}.p^r\left(1-p\right)^{n-r}}\)

\(\color{blue}{P\left(x\right)=C\:\left(n,\:r\right).p^r\left(1-p\right)^{n-r}}\)

Where,

• \(n =\) Total number of events
• \(r =\) Total number of successful events
• \(p =\) Success on a single trial probability
• \(^nC_r=\left[\frac{n!}{r!\left(n-r\right)!}\right]\)
• \(1 – p =\) Failure Probability