Question

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3. Let X have density fx () = 1+1 -1<<1. (a) Compute P(-2 < X <1/2). (b) Find the cumulative distribution Fy(y) and probability density function fy(y) of Y = X? (c) Find probability density function fz() of Z = X1/3 (a) Find the mean and variance of X. (e) Calculate the expected value of Z by (i) evaluating S (x)/x(x)dr for an appropriate function (). (ii) evaluating fz(z)dz, pansion of 1/3 (ii) approximation using an appropriate formula based on Taylor series ex- Solution (1) Calculate the variance of Z and compare it to the approximation using an appropriate formula based on Taylor series expansion of

Answer 1

(a) Given that the pdf for X as fx(x) = *, -1<x<1 lo, elsewhere $0, P(-2<x<3) = s£,(8)dx= [(0)dx + ** dx = 0) (**PETE = (b) (+1,-1<x< 1 and Y = X?. Given that the pdf for X as fx(x) = 2 (0, elsewhere Let Fy is the respective distribution function for Y and f, is the corresponding pdf for Y. Here, we can see that Y = x2 is not one-to-one function. For this, we need to find f, using generic method. As we know, Fy(y) = P(Y Sy) [Y is the random variable, y is the realized value for Y] i.e. Fy(y) = P(X² = y) = P(-Vy sxs y) = Fx(79) - Fx(-Vy................... (1) Therefore, using the relationship between pdf and its distribution function ſi. e. fx(x) = Fx(x)], we can have fyly) = .((y)) = (Fx(Vg) – F&(-V7)) [From (1) i.e. fy(y) = fx(ſy) . (19) – fx(-vy). (-19) ((fx()+fx(-17), if 0 <y<1 .................... (ii) 1 0, ify so = 2/ Using the pdf of X in (ii) we can have (* )_1yif <y<1 fy(y) 1 0, elsewhere

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