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.