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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 X
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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- 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-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]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) = 204I 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 yiii) 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<Van (Y)= E (Y2) - E’(Y) 4I 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 +

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