Random variable. W. is formed as the sum of independent random variables such that W-X+Y where the density functions of
Posted: Thu May 05, 2022 8:21 pm
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.
****PLEASE SOLVE THE PROBLEM SIMILARLY TO THE NOTES. DONT GO OFF TRACK***** Will thumbs up
NOTES TO HELP SOLVE THE PROBLEM:
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: F.(y) 1/a f(x) fx(x) = [u(x) - u(x - a)]: fr(y) = [u(y) − u(y – b)]: 1% a 00 Find fw(w) = fy(v) * fx(x) = = fy(y)fx(w-y) dy y-00 fx (wy) in two steps: 1/a fx(-y) fx(w-y) -a0 This can be animated on the convolution demonstration at: https://phiresky.github.io/convolution-demo/ 1/a w-aw x. y. w axis
Configuration w-aw w-a/w L w-awb w-a bw b w-a (w) Lower Upper w range Limit Limit -- WLD ه ه ن لعلا عما ط له fw(w) overlap T asuch - Jutady * * * - ++ - = - - = 0 ان ۵ ه no = مضاء ط دا مات ما+ ط »
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.
****PLEASE SOLVE THE PROBLEM SIMILARLY TO THE NOTES. DONT GO OFF TRACK***** Will thumbs up
NOTES TO HELP SOLVE THE PROBLEM:
- Random Variable W Is Formed As The Sum Of Independent Random Variables Such That W X Y Where The Density Functions Of 1 (45.33 KiB) Viewed 17 times
- Random Variable W Is Formed As The Sum Of Independent Random Variables Such That W X Y Where The Density Functions Of 2 (55.56 KiB) Viewed 17 times
Configuration w-aw w-a/w L w-awb w-a bw b w-a (w) Lower Upper w range Limit Limit -- WLD ه ه ن لعلا عما ط له fw(w) overlap T asuch - Jutady * * * - ++ - = - - = 0 ان ۵ ه no = مضاء ط دا مات ما+ ط »