$X$ is a random variable uniformly distributed on the real interval [0,1].
Through some experimentation, I found that the probability density function, PDF of:
$X$ is $1$ or $\dfrac{d}{dx}X$
$2X$ is $\frac{1}{2}$ or $\dfrac{d}{dx}X/2$
$3X$ is $\frac{1}{3}$ or $\dfrac{d}{dx}X/3$
$X^2$ is $\frac{1}{2\sqrt{X}}$ or $\dfrac{d}{dx}\sqrt{x}$
$X^3$ is $\frac{1}{3x^{2/3}}$ or $\dfrac{d}{dx}\sqrt[3]{x}$
The PDF is useful in answering questions such as what is the mean of $X^3$ or what is the probability that $0<2x<\frac{1}{21}$?
1) How do I find the PDF of functions in general, something like $X+X^3$?
2) Also, when there is another variable involved, say Y that is a random variable uniformly distributed on the real interval [0,2], how do I find the PDF of expressions like $X+Y^2$ or $XY^2$? This is again most helpful in finding answers like what is the variance of $X+Y^2$ or what is the probability that $XY^2 > 1$?
3) What if X and Y are not uniformly distributed, but follows some continuous distribution like the Poisson or Gaussian? How do I find the PDFs in this case?
Best Answer
You have just discovered that the cumulutative distribution function of an $f(X)$ when $f$ is an invertible monotonuous increasing function can be computed as:
$$\mathbb{P}(f(X)<y)=\mathbb{P}(X<f^{-1}(y)) \; .$$