Question

function, s sin(x) if 0 < x <A otherwise Find the mean and variance of X. Find the mean and variance of the random variable X2 with the fol lowing distributions: - (i) X ~ N(u,0) - (ii) X ~ P(X) - (iii) X ~ Expo(1) "roblem 7 vandam variables,

Answer 1

OW . f(x) s / sina if osx S.A. o eso.w Mean of x is :- . E(x) = Î x fx(x) dx . . T xp xuis in X Sin * *[xfiae dre - Ji (Ssime dou) du?" =*xc-coox) – Seconx) dx]" * Exe Cooke - Sinx ] = * |-* (-1) = sin(a) to + sino] CO

- نر Van (x) = E(x *) - CE (x)) ? Now, E6x9 = F xtsinx du *** [en sinu da). * *[23 S sin da - Jax essin x dra) den] = + [? 6-609x) + 25 x cost de ]," = f[xcosx +ex score de - 2f1(cosx da) di - [ x + x 5inx - 6 mx" = { [x?cosx + ex sinxe + 2 eos x ]* اور ۶ ابر ابر = 1 / -a² cosa +2a sina + 20000] to to - 2.oso = $ [*} – 27 =4632-4) Var (x) = 1 - 2 - 2 ابر

We know that if u ~ N(0, 1) . then jana? A ECU?) = 1 - 0 Now we put U=(x-) where x ~N(M, of) E (X-) = 1 =dif[x? - 2x n + m2] = 4 ► tale [x2] - 2ME(X) + 127=1 → E (x2) = 02 +42 2 2 X१ X n + M2 + 2M

Similarly variance of we is :- Van (02) = 2X1 var (x-4)² = 2 ( Putting U=X-^) =) Var (x-M)² = 204 » Voni (x²_2x" + H2) = 204 van (x- ex ) = 204 (: van(x+C) = Von(x)) > vor (x2) + 412 var(x) - 4 cov (x3, x) = 204 - - Var (x²) + 41²0²_4002 = 204. 1 Covlx,x®) 6v (+*, cou+s)?) = cou couth, 0402+2060 + m2) - =oGov CV, 72) +2M0BCU,U) = 2102] I » var (x²) = 204 +4002 - 4M202

I X E S cu TL v Let; Y= x2 = x=VY If X~ fx (x) Then fy(y) = f (vy) .1J) . where 101= 24 SA i = lry. :: fy (y) = fx(ry) ato y Es' —O

iii) xa Exp(a) x > 0,.230 fx (x) = ne-na fyly) = rent things y yo ,n30 ETA E (X) = var(x) + €2. (X) = 1/2 + 1 h 2 2 . 11. hup Silk ox xh +(0177 hop an to the le - 1 e dy = dz N =24 *** de ci 32" *dz

<Van (Y)= E (Y2) - E’(Y) 4

I we know, | E (xº) - Var(x) + &'(x). Now ) Now, Xa Poi(a) E (x4) = Var(x) + &*(x) var (xx) = E (x4) - E? (x4) = [(x2)) - A +22) 2 = x*fyy) - Cat23) ? 21:0