Question

C2.1 (Probability integral transform.) Let X be a random variable with cu mulative distribution function F, and suppose that F is continuous and strictly increasing on R. (i) Show that F has a well-defined inverse function G : (0,1) → R, which is (ii) Using G, or otherwise, show that the random variable F(X) is uniformly 시 strictly increasing distributed on [0,1

Answer 1

I)
strictly increasing means :
x < y <=> F(x) < F(y)

ii)
we will use the fact that for any A, real :
P( X < A) = F(A)
so
the cdf of h(x) of F(X) is so that :
if x < 0 then h(x) = 0

if x > 1 then h(x) = P( F(X) < 1 < x) = 1
and
if x € [0,1]
h(x) = P( F(X) < x) = P( X < F^-1(x)) = F( F^-1(x))
using the result above for A = F^-1(x)
therefore
h(x) = F( F^-1(x)) = x
therefore
F(X) is uniformly distributed on [0 , 1]