Discrete variables

Uncertain numbers are numbers in respect of which we are in a condition of uncertainty. They can be the number associated with the outcome of a die, to the number which will be read on a scale when a measurement is performed, or to the numerical value of a physics quantity. In the sequel, we will call uncertain numbers also ``random variables'', to come close to what physicists are used to, but one should not think, then, that ``random variables'' are only associated with the outcomes of repeated experiments. Stated simply, to define a random variable $X$ means to find a rule which allows a real number to be related univocally (but not necessarily biunivocal) to an event ($E$). One could write this expression $X(E)$. Discrete variables assume a countable range, finite or not. We shall indicate the variable with $X$ and its numerical realization with $x$; and differently from other notations, the symbol $x$ (in place of $n$ or $k$) is also used for discrete variables.

Here is a list of definitions, properties and notations:

Probability function:

To each possible value of $X$ we associate a degree of belief:

$$f(x)=P(X=x)\,.$$ (4.1)

$f(x)$, being a probability, must satisfy the following properties:

$$0 \leq f(x_i) \leq 1\,,$$ (4.2)

$$P(X = x_i\,\cup\, X = x_j)= f(x_i)+f(x_j)\,,$$ (4.3)

$$\sum_i f(x_i) = 1\,.$$ (4.4)

Cumulative distribution function:

$$F(x_k) \equiv P(X\leq x_k) = \sum_{x_i\leq x_k} f(x_i) \, .$$ (4.5)

Properties:

$$F(-\infty) = 0,$$ (4.6)

$$F(+\infty) = 1,$$ (4.7)

$$F(x_i) - F(x_{i-1}) = f(x_i),$$ (4.8)

$$\lim_{\epsilon \rightarrow o} F(x+\epsilon) = F(x) \hspace{1.0 cm} (right\ side\ continuity)\,.$$ (4.9)

Expectation value (mean):

$$\mu \equiv [X] = \sum_i x_i f(x_i)\,.$$ (4.10)

In general, given a function $g(X)$ of $X$,

$$[g(X)] = \sum_i g(x_i) f(x_i)\,.$$ (4.11)

E$[\cdot]$ is a linear operator:

$$[a X+b] = a [X] + b \,.$$ (4.12)

Variance and standard deviation:

Variance:

$$\sigma ^2 \equiv (X) = [(X-\mu)^2] = [X^2] - \mu ^2 \,.$$ (4.13)

Standard deviation:

$$\sigma = \sqrt{\sigma^2}\,.$$ (4.14)

Transformation properties:

$$(a X+b) = a^2\, (X)\,,$$ (4.15)

$$\sigma(aX+b) = \vert a\vert\,\sigma(X)\,.$$ (4.16)

Binomial distribution:

$X\sim {\cal B}_{n,p}$ (hereafter ``$\sim$'' stands for ``follows''); ${\cal B}_{n,p}$ stands for binomial with parameters $n$ (integer) and $p$ (real):

$$f(x\,\vert\,{\cal B}_{n,p}) = \frac{n!}{(n-x)!\,x!}\, p^x\, (1-p)... ...2, \ldots, \infty \\ 0 \le p \le 1 \\ x = 0, 1, \ldots, n \end{array}\right.\,.$$ (4.17)

Expectation value, standard deviation and variation coefficient:

$$\mu = n\,p,$$ (4.18)

$$\sigma = \sqrt{n\,p\, (1-p)},$$ (4.19)

$$v \equiv \frac{\sigma}{\mu} = \frac{\sqrt{n\,p\,(1-p)}}{n\, p} \propto \frac{1}{\sqrt{n}}\, .$$ (4.20)

$1-p$ is often indicated by $q$.

Poisson distribution:

$X\sim {\cal P}_\lambda$:

$$f(x\,\vert\,{\cal P}_\lambda)=\frac{\lambda^x}{x!}\, e^{-\lambda}... ...y}{l} 0 < \lambda < \infty \\ x = 0, 1, \ldots, \infty\\ \end{array} \right.\,.$$ (4.21)

($x$ is an integer, $\lambda$ is real.)
Expectation value, standard deviation and variation coefficient:

$$\mu = \lambda,$$ (4.22)

$$\sigma = \sqrt{\lambda},$$ (4.23)

$$v = \frac{1}{\sqrt{\lambda}}.$$ (4.24)

Binomial $\rightarrow$ Poisson:

$${\cal B}_{n,p} \xrightarrow [n\rightarrow \lq\lq \infty'' \\ p\rightarrow \lq\lq 0'' \\ (\lambda = n\,p) \\ \cal{P}_\lambda$$