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GATE CSE 2003 | Question: 60, ISRO2007-45

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A program consists of two modules executed sequentially. Let $f_1(t)$ and $f_2(t)$ respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by 

  1. $f_1(t)+f_2(t)$

  2. $\int_0^t f_1(x)f_2(x)dx$

  3. $\int_0^t f_1(x)f_2(t-x)dx$

  4. $\max\{f_1(t),f_2(t)\}$

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5 Answers

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We assume the total time to be ‘t’ units

f1 executes for ‘x’ units.

f1(t) and f2(t) are executed sequentially.
So, f2 is executed for ‘t – x’ units.

We apply convolution on the sum of two independent random variables

f1(t) * f2(t – x) =
  \int_{0}^{t}f_{1}(x)f_{2}(t-x)dx

 
Thus, option (C) is correct.
 

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