# How to find probability density functions?

$X$ 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?

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There numerous useful posts devoted to this problem: 30938, 77873, 55607 and many others. – Sasha Mar 25 '12 at 13:43
To convince you that this answer is for you, let me quote its first words: The simplest and surest way to compute the distribution density or probability of a random variable is often... – Did Mar 25 '12 at 14:21
I apologize but I'm not strong in calculus so I'm still having trouble understanding how to apply your solution in that particular case to the general scenario. Until I do, I'll just leave the question open, hoping someone else can answer it in a different way perhaps. – hollow7 Mar 26 '12 at 23:52

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:
$$\mathbb{P}(f(X)<y)=\mathbb{P}(X<f^{-1}(y)) \; .$$
It is conventional to use capital $F$ for the cumulative distribution function and lower-case $f$ for the density function. And I don't think it's a good idea to use the same letter, capital $X$, to refer both to the random variable and to the argument to the CDF. – Michael Hardy Mar 25 '12 at 14:57
$F_Y(y)=\mathbb{P}(Y<y)$ – Raskolnikov Mar 25 '12 at 14:59