In this problem, your task will be to derive the probability distribution of a t

Question

In this problem, your task will be to derive the probability distribution of a transformed sample Consider a uniform set of samples in [0, 1] and apply function f to each sample (you can assume f is continuous and differentiable) Derive the continuous probability distribution according to which the transformed samples would be distributed. You can assume f is monotonically increasing on the interval [0, 1], Make sure to show your work! You should arrive at an expression for the new distribution (density) function q in terms of values/derivatives/integrals of f. q(x) = .... Use your result from the previous part to show that if f(x) = Squareroot x. then q(x) = 2x.

Explanation / Answer

let

X~U(0,1)

So PDF of X is

g(X)=1

1.

Let f(X)=Y

We have to find the distribution of Y

f(X)=Y

So

X=f-1(Y)

Now

Jacobian=|dX/dY| =|df-1(X)/dY|

So

q(x)=Jacobian*g(f-1(y))

2.

y=x

So x=y2

Now dX/dy = 2y

Hence

Q(X) = 2y*1=2y

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