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Consider a random variable X that always takes a single value, c. I would think that a valid PDF for X would be $f(x) = \begin{cases} 1/c & x = c \\ 0 & \text{Otherwise}\end{cases}$

However, I know the PDF is supposed to be the derivative of the CDF, and in this case derivative of F(x) would be 0, both when $x and when $x\geq c$

What am I missing ?

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    In this case the derivative of F at x would be zero for every x different of c and would not exist at x=c. // Not every random variable has a pdf, for example Dirac ones have not.2011-08-15

3 Answers 3

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What you're looking for is the Dirac delta "function"; specifically

$f(x) = \delta(x-c).$

I've put "function" in scare quotes above, since the Dirac delta is not actually a real function, although it can in many cases be treated as one.

Informally, you can visualize $\delta(x)$ as a function which has an infinitely tall peak at $x=0$ and is zero everywhere else; specifically, the "infinitely tall peak" needs to be just tall enough to have the area under it integrate to $1$. Of course, no actual real-valued function can have such a peak, but it turns out that, if you simply pretend that such a function exists and don't ask any hard questions about what its value at $x=0$ actually is, many calculations will just work as if nothing odd was going on.

Formally, the Dirac delta can be defined as a generalized function, specifically as a distribution. (And no, the similarity of the name with "probability distribution" is not a coincidence.)

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Your $f$ is not a valid PDF, since $ \int_{ - \infty }^\infty {f(x)\,dx} = 0 \neq 1. $

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    No. Since the integral of the zero function is $0$, so is the integral of $f$: changing a function on a set of measure $0$ does not affect the integral.2011-08-15
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A PDF (probability density function) is only for a continuous distribution - more precisely, a distribution that is absolutely continuous with respect to Lebesgue measure. Your distribution is discrete, not continuous, so it does not have a PDF, it has a PMF (probability mass function). There are also singular continuous distributions (but nobody talks about them in elementary courses), and mixtures of the three types.