Probability Density Function
What is a Probability Density Function (PDF)?
The probability density function is the function that defines the density of the probabilities of a continuous random variable over given range. It is denoted by f(x) where, x is the continuous random variable.
How Does a PDF Differ from a PMF?
A PDF is used for continuous random variables, where the probability of any single, exact value is zero. A Probability Mass Function (PMF) is used for discrete random variables, where the probability of specific values can be non-zero.
Write Probability Density Function Formula for Continuous Random Variable X in interval (a, b).
The probability density function formula for the continuous random variable X in interval (a, b):
P (a ≤ X ≤ b) = [Tex]\int\limits^{b}_{a}f(x)dx [/Tex]
What are Necessary Conditions for Probability Density Function?
Necessary conditions for the probability density function are:
- f(x) ≥ 0, ∀ x ∈ R
- f(x) should be piecewise continuous.
- [Tex]\int\limits^{\infin}_{-\infin}f(x)dx = 1 [/Tex]
How to Find Mean of Probability Density Function?
The mean of the probability density function can be calculated by following formula:
E[X] = μ = [Tex]\int\limits^{\infin}_{-\infin}xf(x)dx [/Tex]
Can a PDF Have Negative Values?
No, a PDF cannot have negative values. The value of a PDF for a given point in its domain represents the probability density at that point and must be non-negative.
What Does the Area Under a PDF Represent?
The area under a PDF curve within a certain interval represents the probability that the random variable falls within that interval.
How Do You Find the Mean of a Distribution Using a PDF?
The mean (or expected value) of a distribution using a PDF is found by integrating the product of the variable and its PDF over the entire range of the variable.
What is the Relationship Between PDF and CDF?
The Cumulative Distribution Function (CDF) is the integral of the PDF. It represents the probability that the variable takes a value less than or equal to a certain value.
Can a PDF be Greater Than 1?
Yes, a PDF can be greater than 1 for a narrow interval because it represents probability density, not probability. The total area under the PDF curve over all possible values must equal 1.
How is a PDF Normalized?
A PDF is normalized by ensuring that the integral of the PDF over all possible values of the random variable equals 1. This normalization condition ensures that the PDF correctly represents a probability distribution.
How Do You Calculate the Variance from a PDF?
The variance of a distribution from a PDF is calculated by integrating the square of the difference between the variable and its mean, multiplied by the PDF, over the entire range of the variable.
Probability Density Function
Probability Density Function is the function of probability defined for various distributions of variables and is the less common topic in the study of probability throughout the academic journey of students. However, this function is very useful in many areas of real life such as predicting rainfall, financial modelling such as the stock market, income disparity in social sciences, etc.
This article explores the topic of the Probability Density Function in detail including its definition, condition for existence of this function, as well as various examples.
Table of Content
- What is Probability Density Function?
- Probability Density Function Example
- Probability Density Function Formula
- How to Find Probability from Probability Density Function
- Graph for Probability Density Function
- Probability Density Function Properties
- Mean of Probability Density Function
- Median of Probability Density Function
- Variance Probability Density Function
- Standard Deviation of Probability Density Function
- Probability Density Function Vs Cumulative Distribution Function
- Types of Probability Density Function
- Probability Density Function for Uniform Distribution
- Probability Density Function for Binomial Distribution
- Joint Probability Density Function
- Applications of Probability Density Function
- Solved Examples on Probability Density Function