Question

Let X have a logistic distribution with pdf f(x) = (1 + e-x)2 < r< 0. (1+e-x)2 (a) Show that Y=tex 1+e-x has a U(0,1) distribution. (b) Explain how you can generate random samples of X using uniform random samples over (0,1).

=============

Probability statistics problems

Answer all questions to get UPVOTE

Thanks

Answer 1

Solution :- Given : x has logistic dists with pat Fale- tconco (1+e-x2 a) Let y o u na holten Let gly) be the pdf of Y which is gives by gly) = f(a) dni in 18 ms of dy! - . Consider, y- [ when a='-00, y = 0 ltex bh n=0 , y = 1 e- = II - - 1 = - dog (4) - da dy I (1-4) (1) - Y(-1) 7 4/(1-4) L (1-1) 2. 195 1-yty 7 y L (1-4) 2 dy yo-y)

, 40-4) g(y) = ( y) (V)2 (1-4) 42 42(1-4) is proved. which is pdf of ulo, 1) Y= ulo, 1) altex as follow- 6 To generate ris, from x, first we find calf of X Ex(x) = P[x sa] be SS f(t)dt as (lte-t) 2 (7) 2 I whos t = -00, y = oo t a, y =lten ocy Glte Put y - lte-t 4-1 = e-t ..-t = log (4-1) . t = - Jug (4-)) . dt = -1 dy y-1 • Fx (0) - ten • , r3) = ( 9 - alle 7 - 1 te- y2 Fx (x) - o . Ex (A) = lte-2 1 Iten

As we know that Fx (n) of any r.l. is always ulo) So take F(x) as rie random no, from ulo, (Disth of celf of any r.v. is always u(0,1)) I te- - = l + e-X i e-n = 11 en = 18 . . x = log (1-7) ..13 - dagel The values so obtained are from & values of & are from Vol). logistic dists