Finding the PDF given the CDF

Question

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? — , Jul 19 '10 at 19:31

Paul · Accepted Answer · 2010-07-20T08:52:27.083

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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.

edited Jul 20 '10 at 08:52

answered Jul 19 '10 at 20:15

Paul

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There is a nice online textbook by Pishro-Nik [here](http://www.probabilitycourse.com/chapter4/4_3_2_delta_function.php) showing this more explicitly. – gwr Nov 28 '15 at 11:29
Does something similar hold for multivariate case? (I found the answer [here](http://www.cims.nyu.edu/~cfgranda/pages/DSGA1002_fall16/material/multivariate_random_variables.pdf) page 9). $f(\mathbf x) = \frac{\partial^n F(\mathbf x)}{\partial x_1 \dots \partial x_n}$ – MInner Oct 30 '17 at 15:36
Would you mind give an example that a cdf has a discontinuity? – whnlp Sep 27 '19 at 08:33
@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 $\frac{F(x_2) - F(x_1)}{x_2 - x_1}$? – develarist Dec 05 '20 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 Dec 06 '20 at 01:51

score 12 · Answer 2 · edited Dec 03 '14 at 01:21

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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.

edited Dec 03 '14 at 01:21

guy

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answered Jul 20 '10 at 00:59

seancarmody

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Thats the same as taking the derivative, but just more inaccurate so why would you do it? – Matti Pastell Jul 20 '10 at 09:39
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This would be the approach when the CDF is only approximated empirically. It gives lousy estimates of the PDF, though. – shabbychef Oct 20 '10 at 05:13
Given CDF percentile values, is there a better way to calculate PDF from these discrete values ? – bicepjai Jul 23 '18 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 Oct 02 '18 at 14:54

score 0 · Answer 3 · edited Nov 25 '19 at 03:17

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).

Finding the PDF given the CDF

3 Answers3

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