Finding the PDF given the CDF

How can I find the PDF (probability density function) of a distribution given the CDF (cumulative distribution function)?

• I am not sure I understand the difficulty. If the functional form is known just take the derivative otherwise take differences. Am I missing something here?
– user28
Commented Jul 19, 2010 at 19:31
• I am guessing the question is about multivariate case. Commented Aug 5, 2015 at 23:28

As user28 said in comments above, the pdf is the first derivative of the cdf for a continuous random variable, and the difference for a discrete random variable.

In the continuous case, wherever the cdf has a discontinuity the pdf has an atom. Dirac delta "functions" can be used to represent these atoms.

• There is a nice online textbook by Pishro-Nik here showing this more explicitly.
– gwr
Commented Nov 28, 2015 at 11:29
• Does something similar hold for multivariate case? (I found the answer here page 9). $f(\mathbf x) = \frac{\partial^n F(\mathbf x)}{\partial x_1 \dots \partial x_n}$ Commented Oct 30, 2017 at 15:36
• Would you mind give an example that a cdf has a discontinuity? Commented Sep 27, 2019 at 8:33
• @Paul are you wrong in saying above that the discrete pdf is simply the difference of the cdf, $F(x_2) - F(x_1)$? shouldn't it be $\frac{F(x_2) - F(x_1)}{x_2 - x_1}$? Commented Dec 5, 2020 at 14:25
• @develarist You are correct in that this is imprecise. If the discrete outcomes are consecutive integers, then the difference is sufficient.
– Paul
Commented Dec 6, 2020 at 1:51

Let $F(x)$ denote the cdf; then you can always approximate the pdf of a continuous random variable by calculating $$\frac{F(x_2) - F(x_1)}{x_2 - x_1},$$ where $x_1$ and $x_2$ are on either side of the point where you want to know the pdf and the distance $|x_2 - x_1|$ is small.

• Thats the same as taking the derivative, but just more inaccurate so why would you do it? Commented Jul 20, 2010 at 9:39
• This would be the approach when the CDF is only approximated empirically. It gives lousy estimates of the PDF, though. Commented Oct 20, 2010 at 5:13
• Given CDF percentile values, is there a better way to calculate PDF from these discrete values ? Commented Jul 23, 2018 at 17:41
• In this case, are all x from x1 to xn sorted in an ascending order first so that it is always xn>x(n-1)>x(n-2)>…..x3>x2>x1?
– Eric
Commented Oct 2, 2018 at 14:54

Differentiating the CDF does not always help, consider equation:

 F(x) = (1/4) + ((4x - x*x) / 8)    ...    0 <= x < 2,


Differentiating it you'll get:

((2 - x) / 4)


substituting 0 in it gives value (1/2) which is clearly wrong as P(x = 0) is clearly (1 / 4).

Instead what you should do is calculate difference between F(x) and lim(F(x - h)) as h tends to 0 from positive side of (x).

• (-1) Because this CDF does have a derivative, the random variable is continuous and clearly $P(x=0)=0.$ You appear to be confusing densities with probabilities.
– whuber
Commented Apr 6, 2022 at 14:06