How to Find the Probability Density Function?

The probability density function (PDF) provides the probability distribution of a continuous random variable. It essentially describes the relative likelihood of a random variable taking on a given value. Here's how to understand and find it:

Understanding Probability Density Function

The function f_X(x) provides the probability density at a specific point x. It's important to note that the probability of a continuous random variable taking on any single specific value is zero. Instead, we look at the probability of the random variable falling within a small interval around that value. This probability density is the limit of the probability of a random variable falling into an interval divided by the length of that interval as the interval becomes infinitesimally small. This relationship can be understood as:

Concept: The probability density, f_X(x), at a specific point x represents how concentrated probabilities are around that point. Higher values of f_X(x) suggest that the random variable is more likely to be found in the neighborhood of x.
Calculation: We can calculate this as the derivative of the cumulative distribution function (CDF), F_X(x). The CDF gives the probability that the random variable X is less than or equal to x:

f_X(x) = dF_X(x) / dx = F'_X(x), if F_X(x) is differentiable at x.

Finding the PDF

The key to finding the probability density function is working with the cumulative distribution function (CDF) F_X(x). The CDF is defined as: F_X(x) = P(X ≤ x).

Here is how you can calculate the PDF using the CDF:

Find the CDF, F_X(x): The first step is to determine the CDF of your random variable. This is often provided or can be derived from the definition of your distribution.
Differentiate the CDF: Once you have the CDF, differentiate it with respect to x. The result is the probability density function (PDF)
- Formula: f_X(x) = dF_X(x) / dx

Example

Let's consider a simple example. Suppose the CDF of a random variable X is given by:

F_X(x) =

0, for x < 0
x², for 0 ≤ x ≤ 1
1, for x > 1

To find the PDF, we will differentiate F_X(x) with respect to x in each interval:

*   For x < 0: f<sub>X</sub>(x) = d(0)/dx = 0
*   For 0 ≤ x ≤ 1: f<sub>X</sub>(x) = d(x<sup>2</sup>)/dx = 2x
*   For x > 1: f<sub>X</sub>(x) = d(1)/dx = 0

Therefore, the PDF of X is:

f_X(x) =

0, for x < 0
2x, for 0 ≤ x ≤ 1
0, for x > 1

Key Insights

Non-Negativity: The PDF, f_X(x), is always non-negative. This makes intuitive sense as probabilities can't be negative.
Area Under the Curve: The total area under the PDF curve must equal 1. This represents the total probability of all possible outcomes. This also implies that the probability that X falls within an interval [a,b] is the definite integral of the PDF function from a to b. Specifically, P(a<X≤b) = ∫^b_a f_X(x) dx
Probability within an Interval: Probability of X falling into an interval (x, x+ Δ) = FX(x + Δ) - FX(x) as stated in the reference. This result is very useful, because it allows one to calculate the probability of the random variable falling in a range.

Summary Table

Concept	Description
Probability Density Function (PDF)	f_X(x) = dF_X(x) / dx, provides the density of probabilities of a continuous random variable
Cumulative Distribution Function (CDF)	F_X(x) = P(X ≤ x), provides the probability that the random variable is less than or equal to x
PDF from CDF	Obtain PDF by differentiating CDF with respect to x
Integral of PDF	The integral of PDF within a range a and b provides the probability that X falls in that range.
Total probability	The area under the PDF curve over all possible values is 1.

By finding the CDF and differentiating it, you can effectively find the probability density function, which is a powerful tool to analyze continuous random variables.

askvity