Question

Let x be a continuous random variable having cumulative distribution function f. define the ran- dom variable y by y =f(x). show that y is uniformly distributed over (0, 1).

Answer 1

Given that F is a distribution function then it is going to be said to be non decreasing.

How to define the random variable

Let Y = F(x)

Then we have a distribution function that we can define to be

Gy(Y) = P(Y≤y) = P[F(X)≤y]

=  P[x≤F⁻¹(y)]

The inverse is said to exist given that the value F is non decreasing and it is also said to be continuous.

Such that Gy(Y)  = F[F⁻¹(y)]

then we have to see f to be the distribution of x

Gy(Y) = y

The PDF of Y is denoted as Y = F(X)

then we would have

d/dy [Gy(Y)] = 1

Given that F is a df, then the value of the distribution function would always be 0, 1.

gy(y) would then have to be 1

0 ≤ y ≤ 1

This tells us that F(X) is a uniform variate of 0,1.

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