Question

watables, vere a w ote and A and B e cons. . (20pts.) 3. Let X = Acos Q and Y = Bsind be two random van distributed random variable in the interval (0.71 and (a) Are of X and Y independent? (b) Are X and Y uncorrelated? (c) Are X and Y orthogonal? ca)

Answer 1

6 Given that, X= A cos of and Y= Bsind. where 0 ~ Uniform (on)! (a) x= A cos of - x² = A² cos²d - x² = A² (1-sin o) - (4) Noon Also, y = Bsing - y² B² sine of -) = sind of — (**) B2 2 2 so X and Y are related the above equation So, they are not independent.

(b) To check whether x and y are uncorrelated, it is enough to check whether Cov(x,y) i zero or not. Now, cor (x,y) - Bloerzeeset = E[xy] - E[x] E [y] Now, € [xy] = E [ (Acoso) (Bring] = E AB coso sind ] = A B E [cos o sina] = AB E[ 200s & sind] - A B E [sino 11 11 Eisin2o] I sin 20. I do 2019/12/06 00:05

Let 2d=t 2do = dt - df = dt/ So, I sin 20 do 01 - OL 0 2K , di sint dt/₂ Jo i sint at Il sint at 2 Jo 26 11 l coso - cos 2 J

JULI 191.175/WK 28 So Elxy] =0. Now, E[x] = " A coad do 1 Al cos de -A ( Sing ). - A (0-0) = AXO So, E[x] = 0 -) E[XJE [YJ - OxE[4]=0 a) Com (x,y) = E[xy) - E[XJE [y] = 0-0 So, they are X and Y are uncorrelented. uncoror
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291193 X col shelter and Y (e) To check whether it is are orthogonal, enough it is enough to check whether £[xy]. is zero or not. In answer (b), we saw that E[xy] =0 & X and Y are orthogonal. ms