GATE CS 2003 Q60 — Probability: Random Variables and Distributions

Let $T_1$ and $T_2$ be the continuous random variables representing the execution times of the first and second modules, respectively. Let their probability density functions (PDFs) be $f_1(t)$ and $f_2(t)$. Since execution times are non-negative, we have $f_1(t) = 0$ and $f_2(t) = 0$ for $t < 0$.

Since the two modules are executed sequentially, the total execution time of the program is the sum of the individual execution times:

Assuming the execution times of the two modules are independent, the PDF of the sum of two independent continuous random variables is given by the convolution of their individual PDFs:

We can simplify the limits of integration based on the non-negative nature of the execution times:

• Since $f_1(x) = 0$ for $x < 0$, the lower limit of integration becomes $0$.
• Since $f_2(t - x) = 0$ for $t - x < 0$ (which means $x > t$), the upper limit of integration becomes $t$.

Substituting these limits, we get:

This matches Option C.