I am trying to understand the difference between the two concepts mathematically and also graphically.

Regarding the probability density function (pdf), I know that it shows the probability of a continuous random variable having a value that belongs to an interval. What I don't know is how we make the transition from probability mass function (pmf) to pdf. I would like to know this, as additional knowledge, but it's not the point of my thread, but I'd appreciate links/pdf etc regarding this.

My problem is to understand the difference between pdf and the continuous probability distribution. In fact my problem is understanding the continuous probability distribution.

Example : we have the normal pdf. It's a function that whose value depends on a continuous variable and we can plot it, and we would get the bell shaped graph. So I understand the meaning of pdf, how we express it mathematically and how it looks graphically.

Now I want to know the same about the normal continuous probability distribution.

What it means?

How we express it mathematically?

How it looks graphically?

By knowing the answer to the above 3 questions then I can understand how it's different from the pdf.


1 Answer 1


I suspect what you are calling a "continuous probability distribution" is more commonly called a cumulative distribution function. (If not, I'd be interested in knowing which source seemed to indicate that they are different.)

Indeed, the probability density function $f$ and the cumulative distribution function $F$ are the most important tools for working with continuous random variables. To give the meaning of $F$ (as you've done for $f$), it is simply \begin{equation} F(x) = \mathrm{Pr}(X < x). \end{equation} Mathematically, you can go from one to the other with \begin{align} f(x) &= \frac{d}{dx} F(x) \\ F(x) &= \int_{-\infty}^x f(y) dy \end{align} which are consistent with eachother by the fundamental theorem of calculus. Graphically, if it is still hard to get a sense of the shape from the above formula, you can go to the normal distribution Wikipedia article where both the pdf $f$ and cdf $F$ are plotted. Integrated bell curve from Wikipedia The latter looks like a smoothed out step function. Its minimum and maximum values are 0 and 1 respectively as required for a probability.

  • $\begingroup$ Here i found the term contiuous probability distribution : en.wikipedia.org/wiki/… $\endgroup$
    – imbAF
    Jul 4, 2021 at 14:29
  • $\begingroup$ I think I made a mistake regarding pdf. pdf itself doesn't give your the probability of a variable having a value within a certain range, instead it's the CDF that uses the pdf in the integral with the integral boundaries being the extreme values of the interval from which the variable can take values from. Is this correct? $\endgroup$
    – imbAF
    Jul 4, 2021 at 14:36
  • $\begingroup$ That's right. Regarding "continuous probability distribution", Wikipedia isn't using that term to refer to a function per se. They just want to be able to say that the pdf and cdf are two different formulations of something so they call that something continuous probability distribution. $\endgroup$
    – Connor Behan
    Jul 4, 2021 at 14:41
  • $\begingroup$ One additional thing. If pdf doesn't give you the probability that the variable takes a certain value within an arbitrary range, then for the normal pdf for example, we have the sigmas whose meaning is the probability of the value being withing a certain range around the mean value. Is this infact the CDF? That you can also illustrate it via a graph? Sorry for the way I am expressing myself, but having no background in statistics and jumping to quantum statistical mechanics is a rough thing. $\endgroup$
    – imbAF
    Jul 4, 2021 at 14:47
  • $\begingroup$ For a normal distribution with mean zero, the sigma (standard deviation) is defined by $\sigma^2 = \int_{-\infty}^{\infty} x^2 f(x) dx$ which is not the CDF. However, for a definition in terms of the CDF, you can think of it as the number such that $F(2\sigma) - F(-2\sigma) = 0.95$. $\endgroup$
    – Connor Behan
    Jul 4, 2021 at 15:07

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