Question

Problem 1 Suppose X ~ fx(x), and let Y = aX + b. We know that E(Y) = aE(X) b, and Var(X)a2Var(X). What about the density of Y, fy(y)? Assuming a > 0. Calculate fy(y) using the following two methods (1) Let Fx() P(X x). Calculate Fy(y) = P(Y < y) in terms of Fx. Then calculate fy (2) Calculate Y (y, y + Ay)) Ay fr(y) (3) Give geometric explanations of your result

Answer 1

o xuf(x) 7 Ex(x) = P(x{") and given that ayo Now, YLY <=> XL (Y-b/a So, Fy(Y) = P(YLY) = P(x L 9-b = Ex(y-b) So, by differentiating both sides, we find fyly) á fx (Cy-b)/a). — 27 Now, YLY <=> x < (Y-b/a YLYtoy <=> x < ((utay-b/a So, pl Ye (YY+AY)) = (y kytay) - P(YCY) = P(x< (ytay)- b)- P(x446) = Ex({}+44) b) - Fx (9-b). f() ** (*1994) - Fx ) -

continuous nandom I Variable 37 to Let us define fy(Y)= lim Ply< Y <ytay) Trie sot Toy Here, fyly) gives us the probability density at y; it is the limit of the probability of the interval (y ytay) divided by the length of the interval (Ay) which is going to o. (So, fy (Y) = lim P(y{ y < ytay) aynox oy = geliyor P(<Y+ry) - P(xky) 4y bolo lim Fylytay) - Fyly) d Fyly dy = Fý (y) if Fyly) is differentiable at y. So, geometrically fy (y) is the derivative of Fy (yo) at point Y=y. a s Secant Line v stangent Line fyly) is the limit of the secant line joining P= (y, Fy(y) and a on the graph of Fyly) as Q approaches Pin

By this concept, we can write (A) f(y) = FAX ((ytay-b)/2) - Fx ((-b/a) - Fx ( Yub) sy = la f(y-b) as oy of

For equation A1, we can see from fyly)= Ļ fx (M-b/a), there the variability gets affected by the shift in scale. It is clear from the Jaco bian part (1 x 1 +) and the mean gets affected by both of the shifts in scale and based ( yy-be ie. E(Y)= a E(x) + b both base and scale only scale