Question

- Let X, 1.2.4. U(-1,1) for i € {1, 2, 3). (a) Write down the expected value and variance of X]. Sketch the pdf fxı. [3] (b) Let Y = X1 + X2. Compute the pdf fy of Y and sketch it. Using fy, or otherwise, compute the expected value and variance of Y. (7) (c) Let Z = X1 + X2 + X3 = Y + X3. Using exactly the same technique that you used in part (b), it can be shown that the pdf of Z is fz(2) = 1 (z + 3)2 -22 (z - 3)2 if z < -3or z > 3 if : € (-3,-1] if z € (-1,1] if € (1,3) Note that you do not need to show this yourself. Carefully draw a graph of this pdf - you may use R to do so if you wish. Using fz, or otherwise, compute the expected value and variance of Z. Compute the values of fz(2) and $(2) at z €{-3, -2, -1,0} to two leading decimals, where o() = te-24/2 is the density of a standard normal, and add a sketch of o to your graph. Comment. Hint: you may wish to mention an important theorem in your comment. [5]

Answer 1

ANSWER: Let x; UE-Ll) for if {1,2,3} @ Expected value and variance of XI. EC)H61, 0 & poobability density function of x1 fw.cal): J ISMIST 10. Otherwise palf of xl sketch: b) let y = x1 + x2 -tlg XtRg.. f($1182) : -<4192221 one-one and onto transformation (ali za 1 (u,v). U: Xitx2 0 V = 72 Expressing the old variables of new 21= 4.06 → diaviables, 22: 0 0

The jacoblan of transformation is given by: from 0-d<uze , RUSTI from equation ® <ULL, ICU>1</ -> 7-4 <-usi-u 5) U-15 UCHU föl -cuc- -ICUcitu -3 4-1-2u - Hu o ia for cuco Lu<HU Ut -1 u 0 ut fon ocul U-LCUCI Itu 2 -2 - U- 10 For ICULQ = U-Kual ouTuy

-PLU,V) : 14 f(u): ff (u, v)au , when :-22ULU fluv) DV, when OCUC 2 1), when - GLUCO , when ocu< 2 pop of y: x1+x2 fy: | 4 (2+4) - when -deyco (2-4), when ocyca fyl) Tylus 7 (-2,0) (2.0) Expected value ECY) = Expectand value ecp = tf ongeve you to wyr ysay = $(7449+ + [ga te med *[0=(1-$)+ (4-4).] ECVA) - 0 Cox=ye) ou + dojens - 95oy 4

**(0-("+")65 )] vaniance vy): E (42) -&?Cy). - Ś (C) let z: X+42 +3 = ył X3 Pdf of z is f;(2): 10 in 75-3 Or Z »3 tocz+37 if ze (-3,-1] - gf ze ip zecin] L (3-) 2 if ze (13) 3,00 (-1,0) 1 (0) 0,3 especho value eius veces cuanya aja-09 4 facesto - 4. jezet* 2248 4 24 aj do starenje du. + & -t [4-7425-69-54** 3+ (+2.507 [ú+h-2) +0]

variace cen: ta }(24 tek ne? 20 Jl34 3za ht+ + ]eu-te a sos) de 2 -3 -2 - -Pz(?) 0(2) 0 0.004432 0.0685 0.53991 0.25 0.241975 0.375 0.398 448 s o 62 standard námal table- The distribution of 2 is tending to *** — *** Tronxung you Thanking you ***