%This command provides the text of the last column (Probability)

%This command provides the text of the last column (Probability)
%on page 3
%
%The macro has one parameter:
% 1) The width of the text
%
\newcommand\TThreeProb[1]{%
\parbox[t]{#1}{%
\DisplaySpace{\TThreeDisplaySpace}{\TThreeDisplayShortSpace}

%Formula 9
\TThreeTitle{Normal (Gaussian) distribution:}
\begin{DisplayFormulae}{1}{0pt}{2ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{p(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-(x-\mu)^2/2\sigma^2}},
\Fm{\E[X] = \mu}
\end{DisplayFormulae}

%Formula 10
\TThreeTitle{Continuous distributions:}%
If $\Pr[a<X<b] = \int_{a}^b p(x)\dx$,
then $p$ is the probability density function of $X$.

If $\Pr[X<a] = P(a)$,
then $P$ is the distribution function of $X$.

If $P$ and $p$ both exist then
$P(a) = \int_{-\infty}^a p(x)\dx$.

%Formula 11
\TThreeTitle{Expectation:}
If $X$ is discrete
$\E[g(X)] = \sum_x g(x) \Pr[X=x]$.

\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\SmallChar}{\StyleWithoutNumber}
\unskip
If $X$ continuous then
\def\FirstPart{\E[g(X)]\mbox{}}
\FmPartA{\FirstPart = \int_{-\infty}^{\infty} g(x) p(x)\dx}
\FmPartB{\FirstPart}{= \int_{-\infty}^{\infty} g(x) \, d P(x)}.
\end{DisplayFormulae}

%Formula 12
\TThreeTitle{Variance, standard deviation:}
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{\Var[X] = \E[X^2] - \E[X]^2},
\Fm{\sigma = \sqrt{\Var[X]}}
\end{DisplayFormulae}

%Formula 13
\TThreeTitle{For events $A$ and $B$:}%
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{\Pr[A \Or B] = \Pr[A] + \Pr[B] - \Pr[A \And B]}
\FmPartA{\MathRemark[\relax]{\text{iff $A$ and $B$ are independent:}}}
%Small initial space to show that the remark is only for
%this equation
\FmPartB{xxxx}{\Pr[A \And B] =\Pr[A] \cdot \Pr[B]}
\Fm{\Pr[A \vert B] = \frac{\Pr[A \And B]}{\Pr[B]}}
\end{DisplayFormulae}%

%Formula 14
\TThreeTitle{For random variables $X$ and $Y$:}%
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\FmPartA{\MathRemark[\relax]{\text{if $X$ and $Y$ are independent:}}}
%Small initial space to show that the remark is only for
%this equation
\FmPartB{xxxx}{\E[X \cdot Y] = \E[X] \cdot \E[Y]}
\Fm{\E[X + Y] = \E[X] + \E[Y]}
\Fm{\E[c X] = c \E[X]}
\end{DisplayFormulae}

%Formula 15
\TThreeTitle{Bayes' theorem:}%
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{\Pr[A_i\vert B] =
\frac{\Pr[B\vert A_i] \Pr[A_i]}{\sum_{j=1}^n \Pr[A_j] \Pr[B\vert A_j]}}
\end{DisplayFormulae}

%Formula 16
\TThreeTitle{Inclusion-exclusion:}
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\raggedright
\def\FirstPart{\Pr\Big[\bigvee^n_{i=1} X_i \Big] = \mbox{}}
\FmPartA{\FirstPart \sum^n_{i=1} \Pr[X_i] +}
\FmPartB{\FirstPart}{\sum_{k=2}^n (-1)^{k+1} \sum_{\smash{i_i<\cdots <i_k}}
\Pr\Big[\bigwedge^k_{j=1} X_{i_j}\Big]}
\end{DisplayFormulae}

%Formula 17
\TThreeTitle{Moment inequalities:}
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{\Pr\big[\vert X\vert \geq \lambda \E[X]\big] \leq \frac{1}{\lambda}},
\Fm{\Pr\Big[\big\vert X - \E[X]\big\vert \geq \lambda \cdot \sigma \Big]
\leq \frac{1}{\lambda^2}}
\end{DisplayFormulae}

%Formula 18
\TThreeTitle{Geometric distribution:}%
\begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus 1ex}{\BigChar}{\StyleWithoutNumber}
\Fm{\Pr[X = k] = pq^{k-1}\MathRemark{q = 1-p}},
\Fm{\E[X] = \sum^\infty_{k=1} kpq^{k-1} = \frac{1}{p}}
\end{DisplayFormulae}

\AdjustSpace{2ex plus 1ex minus .5ex}
\noindent
The ``coupon collector'':
We are given a random coupon each day,
and there are $n$ different types of coupons.
The distribution of coupons is uniform.

The expected number of days to pass before we to collect all $n$ types is
$n=H_n$.
}%
}