Let’s compute what \(e^{tA}\) should look like, where \(t\) is some scalar and \(A\) is a diagonalizable matrix. This is a supplement to Second-Order Linear Differential Equations.

Let \(v_1\dots v_{m}\) be the eigenvectors of \(A\). Let \(\lambda_{1}\dots\lambda_{m}\) be the eigenvalues.

Recall that we can therefore diagonalize \(A\) as:

\begin{equation} A = \mqty(v_1& \dots& v_{m})\mqty(\dmat{\lambda_{1}, \dots, \lambda_{m}})\mqty(v_1& \dots& v_{m})^{-1} \end{equation}

read: change of choordinates into the eigenbases, scale by the eigenvalues, then change back to normal choordinates.

A random variable is a quantity that can take on different values, whereby there is a separate probability associated with each value:

• discrete: finite number of values
• continuous: infinitely many possible values

probability mass function

A discrete random variable is encoded as a probability mass function

probability density function

A continuous random variable is represented as a probability density function.

summary statistics

• probability mass function is a description for the random variable: and random variables are usually communicated via probability mass functions
• expected value

adding random variables

“what’s the probability of \(X + Y = n\) with IID \(X\) and \(Y\)?” “what’s the probability of two independent samples from the same exact distribution adding up to \(n\)?”

The Random Walk Hypothesis is a financial econometric hypothesis that stocks have the same distribution and independent of each other: that stocks are a random variable and not predictable in a macro space.

To set up the random walk hypothesis, let’s begin with some time \(t\), an asset return \(r_t\), some time elapsed \(k\), and some future asset return \(r_{t+k}\).

We will create two random variables \(f(r_t)\) and \(g(r_{t+k})\), which \(f\) and \(g\) are arbitrary functions we applied to analyze the return at that time.