Monday, June 24, 2013

Densities and Expectations

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.

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 $M_2(\mathbb R)$.