For the layperson, it's probably most helpful to think of the density function $f$ associated to a random variable $X$ as
the function you integrate to compute probabilities. Similarly, the expected value $E(X)$ is thought of as the
average value that $X$ takes. Often $E(X)$ is defined already in terms of the density function, and it's not clear
to the beginner why $\int_{\mathbb R} x f(x) \ dx$ should compute the expected value of $X$. What should
be perhaps a bit more obvious is that if you integrate $X$ over the entire probability space, with respect to the
given probability measure, then you obtain the average value of $X$. This is indeed how the expectation of $X$
is typically defined in a more analytical setting.

## Monday, June 24, 2013

## Sunday, June 9, 2013

### Complex Numbers as Matrices

Many people take exception with the complex number field $\mathbb C$ because the equality
\begin{equation}\label{eq:neg1}
i^2 = -1
\end{equation}
rankles them in some way. I suspect that the layperson's distaste for this identity is their wont to interpret $i$ as a real number, since they have the most experience with the arithmetic of real numbers. Surely our convention of calling numbers such as $\sqrt 2$, $\pi$, and $e$ real but $-2i$ ``purely imaginary'' doesn't help.

Identities of the form in \eqref{eq:neg1} abound in the wild. For instance, within the ring $M_2(\mathbb R)$ of $2 \times 2$-matrices with real entries, the rotation by $90^{\circ}$ matrix $R$ defined by \begin{equation}\label{eq:matrix} R := \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation} satisfies $R^2 = -\mathbf{1}$. Here $\mathbf{1}$ signifies the identity matrix \[ \mathbf{1}:=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \] which is the multiplicative identity in $M_2(\mathbb R)$.

Identities of the form in \eqref{eq:neg1} abound in the wild. For instance, within the ring $M_2(\mathbb R)$ of $2 \times 2$-matrices with real entries, the rotation by $90^{\circ}$ matrix $R$ defined by \begin{equation}\label{eq:matrix} R := \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation} satisfies $R^2 = -\mathbf{1}$. Here $\mathbf{1}$ signifies the identity matrix \[ \mathbf{1}:=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \] which is the multiplicative identity in $M_2(\mathbb R)$.

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