Question

Let X be a continuous random variable with a strictly increasing cumulative distribution function (CDF) F_X(x) for all x ∈ R. For the random variable F_X(X), find the variance, Var[F_X(X)].

Answer 1

To find the variance of F_X(X), we can use the property that Var[aX] = a^2Var[X] for any constant 'a' and random variable 'X'. In this case, let Y = F_X(X), and we want to find Var(Y). Since F_X(X) has a strictly increasing cumulative distribution function, Y will have a uniform distribution between 0 and 1. The variance of a uniform distribution from 0 to 1 is 1/12.

Step-by-step explanation:

To find the variance, Var[F_X(X)], of the random variable F_X(X), we can use the property that Var[aX] = a^2Var[X] for any constant 'a' and random variable 'X'. In this case, let Y = F_X(X), and we want to find Var(Y).

Since F_X(X) has a strictly increasing cumulative distribution function, Y will have a uniform distribution between 0 and 1. The variance of a uniform distribution from 0 to 1 is given by ((1-0)^2)/12 = 1/12.

Therefore, the variance of F_X(X) is 1/12.