Getting the cumulative distribution function for Sqrt(X) from the cumulative distribution function for X

Question

I've a data set X which consists of randomly generated numbers.

My aim is to plot the cumulative distribution function for square root of X without generating data set for square root of X.

I'm confused and could not think of a solution.

Can somebody let me know how to take the approach here ?

Answer 1

I'll assume the support of the random variable X is nonnegative real numbers since you are taking the square root of it. The cumulative density function (CDF) for X is

F_X(t) = P(X ≤ t)

The CDF for sqrt(X) is

F_sqrt(X)(t) = P(sqrt(X) ≤ t) = P(X ≤ t²) = F_X(t²)

So given the CDF for X, to evaluate the CDF for sqrt(X) at position t, simply evaluate the CDF for X at t².