Understanding the Convolution of Two Random Variables
The convolution of two random variables provides a way to compute the probability distribution of their sum. Here's a step-by-step explanation:
Step 1: Define Random Variables
- Let X and Y be two independent random variables with probability density functions (PDFs) f_X(x) and f_Y(y), respectively.
Step 2: Sum of Random Variables
- We seek the distribution of the sum Z = X + Y. The goal is to find the PDF of Z, denoted as f_Z(z).
Step 3: Convolution Integral
- The PDF of the sum of two independent random variables is given by the convolution of their PDFs:
f_Z(z) = ∫ f_X(x) * f_Y(z - x) dx
Here, z - x represents the value of Y that, when added to X, equals Z.
Step 4: Interpretation of the Integral
- The convolution integral sums the products of the probabilities of X and the shifted probabilities of Y (i.e., Y values that satisfy the sum).
Step 5: Properties of Independence
- Since X and Y are independent, their joint PDF is the product of their individual PDFs:
f_{X,Y}(x,y) = f_X(x) * f_Y(y)
Step 6: Resulting Distribution
- The convolution results in the PDF of the sum Z, confirming that the distribution of the sum of two independent random variables equals their convolution.
Conclusion
- Thus, the sum of two independent random variables is indeed equal to their convolution, providing a fundamental principle in probability theory.