Cumulative Probability Distribution
The Cumulative Probability Distribution for continuous variables is a function that gives the probability that a random variable takes on a value less than or equal to a specified point. It’s denoted as F(x), where x represents a specific value of the random variable. For continuous variables, F(x) is found by integrating the probability density function (pdf) from negative infinity to x. The function ranges from 0 to 1, is non-decreasing, and right-continuous. It’s essential for computing probabilities, determining percentiles, and understanding the behavior of continuous random variables in various fields.
Cumulative Probability Distribution takes value in a continuous range; for example, the range may consist of a set of real numbers. In this case, Cumulative Probability Distribution will take any value from the continuum of real numbers unlike the discrete or some finite value taken in the case of Discrete Probability distribution. Cumulative Probability Distribution is of two types, Continuous Uniform Distribution, and Normal Distribution.
Continuous Uniform Distribution
Continuous Uniform Distribution is described by a density function that is flat and assumes value in a closed interval let’s say [P, Q] such that the probability is uniform in this closed interval. It is represented as f(x; P, Q)
f(x; P, Q) = 1/(Q-P) for P≤x≤Q
f(x; P, Q) = 0; elsewhere
Normal Distribution
Normal Distribution of continuous random variables results in a bell-shaped curve. It is often referred to as Gaussian Distribution on the name of Karl Friedrich Gauss who derived its equation. This curve is frequently used by the meteorological department for rainfall studies. The Normal Distribution of random variable X is given by
n(x; μ, σ) = {1/(√2π)σ}e(-1/2σ^2)(x-μ)^2 for -∞<x<∞
where
- μ is mean
- σ is variance
Normal Distribution Examples
The Normal Distribution Curve can be used to show the distribution of natural events very well. Over the period it has become a favorite choice of statisticians to study natural events. Some of the examples where the Normal Distribution Curve can be used are mentioned below
- Salary of Working Class
- Life Expectancy of human in a Country
- Heights of Male or Female
- The IQ Level of children
- Expenditure of households
Probability Distribution – Function, Formula, Table
A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.
Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.
A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.
In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.
Table of Content
- What is Probability Distribution?
- Probability Distribution Definition
- Random Variables
- Random Variable Definition
- Types of Random Variables in Probability Distribution
- Probability Distribution of a Random Variable
- Probability Distribution Formulas
- Expectation (Mean) and Variance of a Random Variable
- Expectation
- Variance
- Different Types of Probability Distributions
- Discrete Probability Distributions
- Bernoulli Trials and Binomial Distributions
- Binomial Distribution
- Cumulative Probability Distribution
- Probability Distribution Function
- Probability Distribution Table
- Prior Probability
- Posterior Probability
- Solved Questions on Probability Distribution