Conditional PDF (original) (raw)

Last Updated : 23 Jul, 2025

**Conditional Probability Density Function (Conditional PDF) describes the probability distribution of a random variable given that another variable is known to have a specific value. In other words, it provides the likelihood of outcomes for one variable, conditional on the value of another.

Mathematically, for two continuous random variables X and Y, the conditional PDF of X given that Y = y is denoted as:

f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}

Where:

fX|Y(x|y) is the joint probability density function of X and Y.
fY(y) is the marginal probability density function of Y, which is the probability distribution of Y alone.

Here,

**Marginal PDF: f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) dx, which represents the probability distribution of Y regardless of X.
**Conditional PDF: fX∣Y(x∣y) tells us how X is distributed when we know Y is y.

How to Calculate Conditional PDF?

To calculate the **Conditional Probability Density Function (Conditional PDF), we use the relationship between the joint PDF and the marginal PDF and the following steps:

**Step 1: Find the joint PDF fX,Y(x,y). This represents the likelihood of both X and Y occurring simultaneously.
**Step 2: Find the marginal PDF fY(y) by integrating the joint PDF over x: f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx
**Step 3: Calculate the conditional PDF using the formula.

This gives the probability distribution of X given the value of Y=y.

Let’s assume that X and Y have the following joint PDF:

f_{X,Y}(x, y) = 6xy for 0 < x < 1 and 0 < y < 1

**Step 1: Find the marginal PDF of Y:

f_Y(y) = \int_0^1 6xy \, dx = 6y \int_0^1 x \, dx = 6y \left[\frac{x^2}{2}\right]_0^1 = 3y

Step 2: Calculate the conditional PDF of X given Y=y:

f_{X|Y}(x|y) = \frac{f_{X,Y}(x, y)}{f_Y(y)} = \frac{6xy}{3y} = 2x \quad \text{for} \quad 0 < x < 1

Thus, the conditional PDF of X given Y = y is:

f_{X|Y}(x|y) = 2x, \quad 0 < x < 1.

This is how you calculate the conditional PDF.

Properties of Conditional PDF

**Conditional Probability Density Function (Conditional PDF) has several important properties, which are useful in understanding how conditional distributions behave in probability theory and statistics. Here are the key properties:

**Non-Negativity

The conditional PDF must always be non-negative:

f_{X|Y}(x|y) \geq 0 \quad \text{for all} \quad x, y.

This follows from the fact that probability density functions cannot be negative.

**Normalization

The conditional PDF must integrate to 1 with respect to x, given a specific value of y. In other words:

\int_{-\infty}^{\infty} f_{X|Y}(x|y) \, dx = 1 for each fixed y

This ensures that the conditional probability of X given Y = y is a valid probability distribution.

**Conditional Expectation

The conditional expectation of X given Y = y can be computed as:

\mathbb{E}[X | Y = y] = \int_{-\infty}^{\infty} x f_{X|Y}(x|y) \, dx

This is the expected value of X when Y is known to be y.

**Conditional Independence

Two random variables X and Y are conditionally independent given a third random variable Z if:

f_{X,Y|Z}(x, y | z) = f_{X|Z}(x|z) f_{Y|Z}(y|z)

In other words, knowing Z makes X and Y independent. This property is fundamental in areas like graphical models and Bayesian networks.

**Marginalization of Conditional PDF

To obtain the marginal PDF of X, you can integrate out the conditional PDF over the values of Y:

f_X(x) = \int_{-\infty}^{\infty} f_{X|Y}(x|y) f_Y(y) \, dy

This shows how the marginal PDF of X can be recovered from the conditional PDF and the marginal PDF of Y.

**Conditional CDF

The **conditional cumulative distribution function (CDF) of X given Y = y is related to the conditional PDF by:

F_{X|Y}(x|y) = \int_{-\infty}^{x} f_{X|Y}(t|y) \, dt.

This gives the probability that X is less than or equal to X, given that Y = y

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