What is a Conditional PDF?
A Conditional Probability Density Function (Conditional PDF) tells us how likely a value of one random variable is, given that we already know the value of another variable.
Think of it like this: You’re picking a student randomly, but you know they are from class 10. Now, what’s the chance their height is between 5ft and 5.5ft? That’s what a conditional PDF helps calculate.
Formula:
fX|Y(x|y) = fX,Y(x, y) / fY(y), where fY(y) ≠ 0
This means: Divide the joint probability by the probability of what you already know.
What You Need
To use the conditional PDF formula, you need two things:
- Joint PDF (fX,Y(x, y)): This gives the probability of both X and Y happening together.
- Marginal PDF (fY(y)): This is the probability of just Y happening, no matter what X is.
Easy Example
Let's say:
fX,Y(x, y) = 2 for 0 ≤ x ≤ y ≤ 1
Step 1: Find fY(y) = ∫0y 2 dx = 2y
Step 2: Use the formula:
fX|Y(x|y) = fX,Y(x, y) / fY(y) = 2 / 2y = 1/y, for 0 ≤ x ≤ y
So, if Y = 0.5, the conditional PDF is 1/0.5 = 2 for x between 0 and 0.5.
Important Points
- A conditional PDF is always ≥ 0
- The total area under the conditional PDF (for fixed Y) is 1
- To find a probability like P(a < X ≤ b | Y = y), you integrate:
∫ab fX|Y(x|y) dx