# Hw9 - 4.3.6(AK(a By the computational formula Cov(X ltrl.ty...

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4.3.6. (AK) (a) By the computational formula, Cov(X, y) : EIX.lf] _ ltrl.ty. We can compute p,t : (*Xo) + (ixl) + (ix-l) :0 E[xYl : lrJq .rt):.r(ixt'(-t)) + (1x1 '1) + (ix2'0) Therefore, Cov(X, Y) : O - 0 : 0. Consider the probabiliy p[X: Oly : ol. plx :oly : oJ = rtxj,o' r=: ol : t/aP[Y : O] : ll, :r. !9wever,P|x:0]=i.Therefore,P|x:0|y=o|+Pw:0],hence X and Y are not independent E[xi] =1, t*(tr) : 1(r - 11 - n- I n z?. n, n2 Also, E[XjXkl:t.PlXi:1,Xk - U: * *, hence cov(xi,x,): * * _* :: ;;#T. Therefore we can compute Etxt :Eti&l =i E[xil=i]:,.1=r, i=l d=l d=l'o r, and ,l var(x) = var(! &) : d=l !v*1a)+! D cov(x5,x,) d:l f, ir-1 t-+ a^2 )"';'+I \- ' A no u rft.,n2(n_l) 1 sf/47 t{b (b) Since X has the uniform distribution on (-1,1), the p.d.i of X hastheconstant^*rt'r"n":'r,.,',.,:qA=1-'lr,ilil'Llr'wt) Erxt= !,i"*:irl,rda):;(;-i) :o \$rr, X. A ,il l* B) ar and .-^ P\nAtY. pJ'' P(rGA) E[xyr:.alxs] :'[ :,,*=:(:-]):0. : p(7a E) = " l, /brl h.r;f*M f A ( p y-rd ay?,,t +' t" I on- rtY 4.9.19. Let x;= 1 or 0 respectirrely if person i gets their own gift. Then x = Pl=r x; is the number of_peopre who pick the same gift that they brought' Since each Xr has the Bernoulti distribution with Ju".ur" proba- bilily I/ni [email protected] -..

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