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I am currently reading through Elementary Stochastic Calculus with Finance in View and I am going through the probability review section.

Intuitively I know the difference between a Probability Mass Function and a Cumulative Distribution Function. However when discussing discrete distributions the author uses funny language:

$F_X(x) = \sum_\limits{k: x_k \le x} p_k, x \in \!R$

Where $0 \le p_k \le 1$ $\forall k$ and $\sum p_k = 1$

Intuitively this makes sense. This distribution function seems to be the Cumulative Distribution Function. He goes on to describe other distributions such as poisson, etc.

Now I understand the PMF/CDF difference. However the author uses terms like distribution function to describe what I would consider the CDF, and the term distribution to describe what I would consider the PMF.

Is this older terminology for these things? Or are distribution and distribution function referring to something separate? Its even more confusing because its obvious from the above the distribution function is a function, but the author refers to the below as a "distribution" (for example):

$n\choose{k}$ $p^k(1-p)^{n-k}$

Which is also a function!

Should I just change the terminology such that:

  1. Cumulative Distribution Function = Distribution Function
  2. Probability Mass Function = Distribution

Or am I missing something here?

Thanks!

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    $\begingroup$ That usage is common, though 'distribution' to mean density or pmf isn't universal $\endgroup$ – Glen_b -Reinstate Monica Aug 5 '18 at 12:03
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    $\begingroup$ A distribution is a mathematical object distinct from any particular description of it. Common forms of description of a discrete distribution include the CDF, PMF, characteristic function, and cumulant generating function. This is somewhat akin to the distinction between where you live (as a specific plot of land or apartment) and its address, which could be latitude-longitude coordinates, a ZIP+4 number (in the US), State Plane coordinates (in the US), or whatever. It is usual to speak of both the address and the thing it denotes as a "location": likewise for distributions. $\endgroup$ – whuber Aug 5 '18 at 14:22
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Distribution function usually means Cumulative distribution function (CDF), so there is nothing for you to change the terminology. When it is clear from context, the word 'cumulative' is often dropped. As for the other question, both the CDF and PMF can be used to specify the distribution of a discrete random variable.

Probability distribution is the distribution of total probability over a partition $S$, support of the random variable $X$. In particular, if $X$ is a discrete random variable, then $S$ is countable.

A probability distribution is measured by the (cumulative) distribution function $F(x)$ defined by $$F(x)=P\{\omega:X(\omega)\leqslant x\}$$, which we simply write as $$F(x)=P(X\leqslant x)$$

Suppose $X$ is a discrete random variable having distribution function $F$.

Define

\begin{align} p(x)&=\text{ amount of jump of }F\text{ at }x \\&=F(x)-F(x-0)\qquad,\text{ where }F(x-0)=\lim_{h\downarrow0}F(x-h) \end{align}

Note that $p:\mathbb R\to \mathbb R \,(\text{ or }\mathbb R\to [0,1] )$ such that

  • $p(x)\geqslant 0\qquad\qquad\qquad\qquad\qquad\quad,\text{ since $F$ is non-decreasing }$

  • $\displaystyle\sum_x p(x)=F(\infty)-F(-\infty)=1\quad,\text{ due to telescopic sum }$

Here $p$ is called the probability mass function (PMF) of the random variable $X$.

So there is a one-to-one correspondence between $p$ and $F$. That is, $p$ can be defined in terms of $F$ as seen above, and similarly $F$ can be defined in terms of $p$:

$$F(x)=P(X\leqslant x)=\sum_{j:j\leqslant x}p(j)$$

Hence both the PMF and CDF are measures of the distribution of $X$.

But you are right that something like $p(x)=\binom{n}{x}p^x(1-p)^{n-x}$ is not the distribution itself but the probability mass function of a certain distribution. The function $p(x)$ specifies the distribution of $X$. More precisely, one should say $$p(x)=\begin{cases}\binom{n}{x}p^x(1-p)^{n-x}&,\text{ if }x=0,1,\cdots,n\\0&,\text{ otherwise }\end{cases}$$ is the PMF of the random variable $X$, indicating the support of $X$ as well as the parameter space ( here, $0<p<1$ and $n$ is a positive integer). Of course, the distribution of $X$ is given by $p(x)$, so that is what the author meant although he/she should have been explicit.

There are situations where it is easier to write down the CDF than the PMF/PDF and vice-versa. But both are valid to denote the probability distribution of a random variable.

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  • $\begingroup$ I had no idea they were interchangeable! It makes sense now that you can write either to describe the distribution because CDFs and PDFs are related. Thanks so much for this. $\endgroup$ – lolo Aug 5 '18 at 19:26
  • $\begingroup$ I should have also mentioned that distribution is not restricted to just PMF/PDF and CDF. Moment generating functions, probability generating functions, characteristic functions etc. are also valid means of denoting a distribution. $\endgroup$ – StubbornAtom Aug 5 '18 at 19:34
  • $\begingroup$ So, if I'm understanding this correctly, a PMF/PDF, CDF, moment generating function, etc are probability measures on our distribution? $\endgroup$ – lolo Aug 5 '18 at 19:44
  • $\begingroup$ @lolo Let's just keep it simple by saying that all of them describe the probability distribution. $\endgroup$ – StubbornAtom Aug 5 '18 at 19:50
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Some time ago, I wrote an answer to a similar question at the end of which I said

While there might be a very restrictive definition of the phrase probability distribution that some people insist on, the colloquial use of the term broadly encompasses the pdf and the CDF and the pmf (probability mass function which is also called the ddf or discrete density function) and whatever else we might want to include as descriptive of the probabilistic behavior of a random variable. For example, the phrase

the probability distribution of $X$ is uniform on $(a,b)$

will hardly ever be interpreted as meaning that the CDF of $X$ has constant value on $(a,b)~$!! Although it is the distribution which is alleged to be uniform, everyone in his/her right mind will take that as meaning that the density of $X$ has constant value $(b-a)^{-1}$ on the interval $(a,b)$ (and has value $0$ elsewhere). Similarly, for "$X$ is uniformly distributed on $(a,b)$" when what is meant is that the pdf of $X$ has constant value on $(a,b)$.

As another instance of colloquial usage of distribution to mean density, consider this quote from a recently posted answer by Moderator Glen_b.

"Saying the mode implies that the distribution has one and only one."

A density might possess a unique mode but a CDF cannot have a unique mode (in the unextended reals). However, no one reading that quote is likely to think that Glen_b meant the CDF when he wrote "distribution".

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