Using a (nonlinear) transformation we can transform any continuous distribution into a normal distribution. To see this, if g is the normalizing transformation for a continuous random variable with density f(x) and be the density of the standard normal distribution, then it satisfies the functional equation (x) = g'(x)f(g(x)). This means that g obeys the ordinary differential equation g' (x) = (x) / f(g(x)) and the inverse function g ¹ transforms the original continuous variable into the normal distribution¹. Let (x) = integration of (z)dz over limits - to x . Now consider the following statements.

I. ¹ transforms Uniform(0, 1) into standard normal.

II. ¹ (1 e ^X) transforms the exponential distribution e ^x for x > 0 into standard normal. Which of the above statement(s) is/are correct?

(A) Only I.

(B) Only II.

(C) Only I and II.

(D) Neither I nor II.

¹This fact is useful for confidence interval and hypothesis testing where many statistical techniques are developed for normal distribution. To deal with the normality assumption, we can transform the distribution to normal, build a confidence interval, and then transform back to get the confidence interval on the original scale with the same confidence level.