Calculate CDF from PDF

How to Calculate CDF from PDF

To calculate the Cumulative Distribution Function (CDF) from the Probability Density Function (PDF) for a continuous random variable, integrate the PDF. Use the formula F(x) = ∫[−∞,x] f(t) dt, where f represents the PDF and x the value up to which the probability is calculated. This process yields the CDF, F(x), which measures the probability that the random variable takes a value less than or equal to x.

Steps for Calculating CDF

Begin by identifying the PDF of the continuous random variable, represented as f(t). Proceed by integrating this function from negative infinity to x. This integral calculation quantifies the accumulated probability and thus, constructs the CDF, F(x).

Relationship Between CDF and PDF

The CDF F(x) can be derived by integrating the PDF f(t). Conversely, the PDF can be found by differentiating the CDF. This integral and differential relationship ensures that the PDF accurately represents the rate of change of probabilities detailed by the CDF.

Understanding and applying the integral process for CDF calculation is crucial for accurately analyzing the distribution and probabilities of continuous random variables.

How to Calculate CDF from PDF

Understanding how to calculate the Cumulative Distribution Function (CDF) from the Probability Density Function (PDF) is essential for statistical analysis involving continuous random variables. This section provides a concise guide on performing this calculation efficiently.

Step-by-Step Guide

To derive the CDF from a given PDF, one needs to integrate the PDF over the desired range. The formula to calculate the CDF, denoted as F(x), from the PDF, denoted as f(x), is given by F(x) = \int_{-\infty}^x f(t) \, dt. This fundamental relationship underscores that the CDF at any point x represents the area under the PDF curve from -∞ to x.

Normalization Condition

For the calculation to be valid, the total area under the PDF must equal one. This condition ensures that the PDF correctly represents the probability distribution of a random variable.

Example Application

Consider a scenario where a random variable X represents the time, in minutes, a person waits for an elevator. The maximum wait time is 2 minutes. The PDF in this case might be defined as follows:f(x) = 0 for x < 0, f(x) = x for 0 \leq x \leq 1, f(x) = 2-x for 1 < x \leq 2, and f(x) = 0 otherwise. To find, for example, F(0.5), compute F(0.5) = \int_{0}^{0.5} t \, dt = 0.125, and similarly, F(1.5) can be computed by F(1.5) = \int_0^1 t \, dt + \int_1^{1.5} (2-t) \, dt = 0.875.

Deriving PDF from CDF

If necessary, the original PDF can be retrieved by differentiating the CDF. This operation confirms that the PDF and CDF are fundamental inverses in distribution function calculations.

This concise explanation provides clear guidelines on how to compute the CDF from the PDF for any continuous random variable, reinforcing fundamental probabilistic concepts in practical applications.

Calculating CDF from PDF: Practical Examples

Example 1: Uniform Distribution

Consider a uniform distribution with a PDF defined as f(x) = 1 for 0 \leq x \leq 1. To find the CDF, integrate the PDF from 0 to x. Calculating gives F(x) = \int_0^x 1 \, dx = x. The CDF F(x) is then x for 0 \leq x \leq 1.

Example 2: Exponential Distribution

For an exponential distribution with rate parameter \lambda, the PDF is f(x) = \lambda e^{-\lambda x} for x \geq 0. Integrating from 0 to x gives the CDF: F(x) = 1 - e^{-\lambda x}. This example highlights the typical behavior of exponential distributions where F(x) increases rapidly initially, then slows as x increases.

Example 3: Normal Distribution

The normal distribution PDF is characterized by f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, where \mu is the mean and \sigma the standard deviation. The CDF cannot be expressed in elementary functions and typically requires numerical integration or the use of tables like the Z-table for standard normal distribution.

Example 4: Beta Distribution

Consider a beta distribution with parameters \alpha and \beta. The PDF is f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1} for 0 \leq x \leq 1. The CDF is typically evaluated through numerical methods due to the complexity of the integral, especially for non-integer values of \alpha and \beta.

Example 5: Triangular Distribution

For a triangular distribution ranging from a to b with a peak at c, the PDF is piecewise-defined. Calculating the CDF requires integrating these piecewise functions, giving a piecewise CDF that increases linearly from a to c and then decreases from c to b.

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