Why the integral of the transformed probability density function is not equal to 1.

Question

Hello everyone There was a question when solving the task. I have been given a random variable whose probability density function obeys the normal distribution law with parameters m = 2, d = 1.5. This is how my function looks like. I am also given a conversion function $g(x) = \sqrt{x}$. According to the assignment, I need to find the distribution function and the probability density function and the distribution function of the transformed random variable. I kind of found this function using the functional transformation of random variables, but I am haunted by the fact that its integral is not equal to 1, but is equal to 0.949. So my distribution function is not equal to 1 at infinity. Help me figure it out, please. Thank you all!

Answer 1

Your original distribution on $x$ is normal, which means that $x$ can be negative. Since $g(x) = \sqrt{x}$ doesn't yield a real number when $x<0$, the transformed distribution is undefined.

I also notice that the number you get as the integral of your transformed distribution, 0.949, is the same value you get when integrating your original normal distribution from $0$ to $\infty$. This underlines the fact that you're not taking into account values of $x$ less than $0$.