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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:

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

$ f(x)$, being a probability, must satisfy the following properties:
    $\displaystyle 0 \leq f(x_i) \leq 1\,,$ (4.2)
    $\displaystyle P(X = x_i\,\cup\, X = x_j)= f(x_i)+f(x_j)\,,$ (4.3)
    $\displaystyle \sum_i f(x_i) = 1\,.$ (4.4)

Cumulative distribution function:

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

Properties:
    $\displaystyle F(-\infty) = 0,$ (4.6)
    $\displaystyle F(+\infty) = 1,$ (4.7)
    $\displaystyle F(x_i) - F(x_{i-1}) = f(x_i),$ (4.8)
    $\displaystyle \lim_{\epsilon \rightarrow o} F(x+\epsilon) = F(x)
\hspace{1.0 cm} (right\ side\ continuity)\,.$ (4.9)

Expectation value (mean):

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

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

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

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

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

Variance and standard deviation:

Variance:

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

Standard deviation:

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

Transformation properties:
Var$\displaystyle (a X+b)$ $\displaystyle =$ $\displaystyle a^2\,$   Var$\displaystyle (X)\,,$ (4.15)
$\displaystyle \sigma(aX+b)$ $\displaystyle =$ $\displaystyle \vert a\vert\,\sigma(X)\,.$ (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):

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

Expectation value, standard deviation and variation coefficient:
$\displaystyle \mu$ $\displaystyle =$ $\displaystyle n\,p,$ (4.18)
$\displaystyle \sigma$ $\displaystyle =$ $\displaystyle \sqrt{n\,p\, (1-p)},$ (4.19)
$\displaystyle v \equiv \frac{\sigma}{\mu}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{n\,p\,(1-p)}}{n\, p} \propto \frac{1}{\sqrt{n}}\, .$ (4.20)

$ 1-p$ is often indicated by $ q$.
Poisson distribution:

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

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

($ x$ is an integer, $ \lambda$ is real.)
Expectation value, standard deviation and variation coefficient:
$\displaystyle \mu$ $\displaystyle =$ $\displaystyle \lambda,$ (4.22)
$\displaystyle \sigma$ $\displaystyle =$ $\displaystyle \sqrt{\lambda},$ (4.23)
$\displaystyle v$ $\displaystyle =$ $\displaystyle \frac{1}{\sqrt{\lambda}}.$ (4.24)

Binomial $ \rightarrow$ Poisson:

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


next up previous contents
Next: Continuous variables: probability and Up: Random variables Previous: Random variables   Contents
Giulio D'Agostini 2003-05-15