# Transformation of probability density function

• Mar 23rd 2010, 08:36 PM
My Little Pony
Transformation of probability density function
Let X be a continuous random variable with a strictly increasing (and therefore 1-1) cumulative distribution function F(x) = P [X <= x] and probability density function f(x).

(a) [6 marks] What is the probability density function of Y = F(X)?
(b) [2 marks] Evaluate P [Y <= 0.5].

So far, I got that:

F(y) = P[Y <= y] = P[F(X) <= y] = P[X <= F^-1(y)]

I'm not sure how to proceed from there?
• Mar 23rd 2010, 09:03 PM
mr fantastic
Quote:

Originally Posted by My Little Pony
Let X be a continuous random variable with a strictly increasing (and therefore 1-1) cumulative distribution function F(x) = P [X <= x] and probability density function f(x).

(a) [6 marks] What is the probability density function of Y = F(X)?
(b) [2 marks] Evaluate P [Y <= 0.5].

So far, I got that:

F(y) = P[Y <= y] = P[F(X) <= y] = P[X <= F^-1(y)]

I'm not sure how to proceed from there?

• Mar 24th 2010, 12:18 AM
matheagle
Quote:

Originally Posted by My Little Pony
Let X be a continuous random variable with a strictly increasing (and therefore 1-1) cumulative distribution function F(x) = P [X <= x] and probability density function f(x).

(a) [6 marks] What is the probability density function of Y = F(X)?
(b) [2 marks] Evaluate P [Y <= 0.5].

So far, I got that:

F(y) = P[Y <= y] = P[F(X) <= y] = P[X <= F^-1(y)]=F(F^-1(y))=y

which is a UNIFORM (0,1)