Question

Question 13 The cumulative distribution function of X is given by Fx (x) = {-kr <0 0<x<2 > 2 Find (a) the value of k, (b) the probability density function fx (x), (c) the median of X, (d) the variance of X.

Answer 1

An as below:- Given cor for x en po es xco fule) = x -ke; ocasa l ' a22 a now consider, fol2)=1 x - ke2 = 21 2-k (25 = 1 2-4k-1 D 2-1 = 4K = 4k = K - 20.25 o jaso f(a)-3-toas os 252 6 now consider PDF of, we know that PDF of x = f(x) = f(x) = a (a-kx²) t = 1-2kn 2kn 1-2kxos xc 2 PDF of x = f (x) = 2 elsewhere.

we flow consider, from (Df of x; let am be the median then know that, fx(mm) - -0.5 em - k(m) 0.5 xom 1-kam] =0.5 & kau – im to.5 = o. – akik – 24m+1=0. + 2k renn – Xm – Alm +1=0 wam +226 to The roots above Eqr are Am = 3.4142 (8) 0.5858 Since here, 3.4142 & Co, 2) Am = 0.5858 & Medians - an 20.5858 & now consider from part 6 E(x) = Jetbelde - Ja (1-2kx) de - Sen -2 ka?)dx = ( - 2kqy?

Now consider, 86-2 (2_0%) = -20) = 2-) (83-2 - $ (64-5) = 6* = 0.6667 - [E6%)= 0.6667 |_o 662 = jmf məclu - 11+(1-2km) om -fee_skaidru - The ? FB - SE669-)(R6*) - Obce -2-84 E z = 0.6667 TE(x²) = 0.6667 low consider from ears ♡ & , Vaciance ofx, v(x) = E(X2) – [E(x)] - - 0.6667 -0.6667 A ...16-efde