Probability distribution of $e^X$, when $X ~$ uniform$(0,1)$ What is the probability distribution of $e^X$, when $X$ is a random variable and follows the uniform distribution $U(0,1)$?
I noticed that the probability of $e^X$ decays exponentially from $1$ to $e$, even though probability distribution of $U(0,1)$ is constant. Is there an explanation why this is so? Clearly, each number is as likely to be picked, and thus shouldn't each result be equally likely?

Call $f(x)$ the probability distribution function of the random variable $X$. For a transformation of the form $y = y(x)$ the number of events in a given interval has to be the same regardless of the name you give to the variable
$$
f(x)dx = f(y)dy
$$
or in other words
$$
f(y) = f(x)\left|\frac{dx}{dy} \right|
$$
where the absolute value is there to ensure that the pdf is positive. For $y=e^x$, we have $x = \ln y$ and then $dx/dy = 1/y$. Moreover $f(x) = 1$ because $X\sim u(0,1)$, the result is then
$$
f(y) = 1/y
$$
Here's a small simulation after sampling $x$ and generating $y=e^x$. The dashed line is just $f(y)=1/y$

