Random variable. W. is formed as the sum of independent random variables such that W-X+Y where the density functions of X and Y are fx(x)=(1/a)[u(x)-
u(x-a)] and fy(y)=bu(y)Exp(-by) for a>O and b>O. Determine:
a) fw(w) Ans: fw(w) = 0 for w<0, fw(w) = (1-e^ (-bw))/a for O<w<a, fw(w) = (1/a)e^-bw)[e^(ab)-1]
b) Sketch fw(w).
Note: You must draw sketches of fx(-v). fx(w-y), and the overlap of fy(y) and fx/w-y) indicating the appropriate limits of integration for each relevant range.
Otherwise you can claim only half credit.
Random Variable W Is Formed As The Sum Of Independent Random Variables Such That W X Y Where The Density Functions Of 1 (43.28 KiB) Viewed 28 times
Random Variable W Is Formed As The Sum Of Independent Random Variables Such That W X Y Where The Density Functions Of 2 (57.49 KiB) Viewed 28 times
4) Example 4.6-1-Piecewise Convolution. This is similar to the first example except that uniform distributions are used (instead of saw-tooths) in order to simplify integration. Determine the density function fw(w) given that W=X+Y where X and Y are independent random variables having uniform density functions: Fr(y) 1/a f(x) fx(x) = [u(x) - u(x - a)]: fr(y)=[u(y)-u(y- b)]: 1/b Find fw (w) = f(y) * fx(x) = fr (v) fx (w - y) dy fx (w - y) in two steps: 1/a fx (w-y) 10 -a0 This can be animated on the convolution demonstration at: https://phiresky.github.io/convolution-demo/ y-00 1/a w-a w x, y, w axis
Configuration W-a (w w-a/w 16 w-awb w-a bw b w-a (w) Lower Upper w range Limit Limit 13 O WLD overlap. O W OL Wha ab wa алось asuch ju they E L bawat wa b = √5 + 2 dy=[12] 6₁ =b-(wa) 9+6w 46 b 2 8 حال له fw(w) O no O مات
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