Probability Density Function of Product of R.V.

Stochastic Process

Laws of large numbers (LLN's) specify what 'become deterministic' means.
They only operate within the extended model, in other words, laws of large numbers don't apply to the real world

Let's talk about Markov bounds, chebychev bounds and chernoff bounds. what it says is if Y is a non-negative R.V. with an expectation E[Y], then for any real y greater than 0, the probability that Y is greater than or equal to y is less than or equal to the expected value of Y divided by y. The proof of it is by picture.

Properties of Limit Inferior & Limit Superior

Lebesgue's Integrability Condition

Lebesgue's integrability condition, aka Lebesgue's criterion for Riemann integrability, Riemann--Lebesgue theorem

A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous a.e.

The criterion has nothing to do with the Lebesgue integral. It is due to Lebesgue and uses his measure zero, but makes use of neither Lebesgue's general measure or integral.

Almost Everywhere (a.e.) & Almost Surely (a.s.)

almost everywhere:  used in measure theory.
almost surely:  used in probability theory.

Call Option

A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation, to buy an agreed quantity of a particular commodity or financial instrument (the underlying) from the seller of the option at a certain time (the expiration date) for a certain price (the strike price). The seller (or "writer") is obligated to sell the commodity or financial instrument to the buyer if the buyer so decides. The buyer pays a fee (called a premium) for this right.

Risk Aversion

In economics and finance, risk aversion is the behavior of humans (especially consumers and investors), when exposed to uncertainty, to attempt to reduce that uncertainty. It is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. For example, a risk-averse investor might choose to put his or her money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.