Question

In the Program Evaluation and Review Technique (PERT), we are interested in the total time to complete a project that is comprised of a large number of

subprojects. For illustration, let X1, X2, X3 be three independent random times for

three subprojects. If these subprojects are in series (the first one must be completed

before the second starts, etc.), then we are interested in the sum Y = X1 +X2+X3.

If these are in parallel (can be worked on simultaneously), then we are interested in

Z = max(X1, X2, X3). In the case each of these random variables has the uniform

distribution with pdf f(x)=1, 0 <x< 1, zero elsewhere, find (a) the pdf of Y

and (b) the pdf of Z.

How do we get the piecewise function for the results of (a)??

Done Step 1 of2 ou ou CX ονον fur (UV) The joint probability density function ofis, u-x2 y U+X, is, s 1S, Similarly, the probability density function ofO+i elsewhere

Answer 1

(a) U = X1 + X2 The pdf of U is where, fx,41) = 1 if 0 < x1 < 1 = 0 otherwise = 0 otherwise Since 0< u< 2 so for 0< u < 1 then 0 < 1< u and when 1 <u < 2 then u -1< r1 <1 Hence the pdf of U is 0 4 -0 otherwise Now take Y = + X3 then the pdf of Y is fy(y) = where, fx,(x) = 1 if 0 < x3 < 1 = 0 otherwise = 0 otherwise

Hence the pdf of Y = X1+ X2+ X3 is 0 0 (3 - y)2 y-2 0 otherwise (b) CDF of Z is = POG < z)P(X2 < z)P(X3 < z) since X1, X2, X3 are independent PDF of Z is dFz(z).2 fz(2)- 0 otherwise.