Applied Probability

Read these first

1. 01Little's Law

   A strikingly simple relationship, the average number of items in a system equals the average rate items arrive times the average time each one spends there, that holds for almost any queue or system, from an order book to a coffee shop line, regardless of the details of how arrivals or service times are distributed.

2. 02The Cramér-Lundberg Ruin Model

   A model of an insurer's (or a trading desk's) capital cushion steadily growing from premiums or carry but taking random hits from claims or drawdowns, and the mathematics of how likely that cushion is to ever hit zero.

3. 03The M/M/1 Queue and Waiting-Time Distributions

   A single-server queue with random arrivals and random service times has a wait time that doesn't grow smoothly as the server gets busier, it blows up, and the M/M/1 model is the cleanest way to see exactly how.

4. 04The Renewal Reward Theorem

   A result about processes that repeat in cycles, each cycle earning some reward and taking some random length of time, showing that the long-run average reward per unit time simply equals the average reward per cycle divided by the average cycle length.

5. 05The Snell Envelope

   The mathematical object that tells you, at every moment, exactly how much a decision you could walk away from right now is worth, computed by working backward from the last possible moment.

Then the rest

1. 06Hazard Rate and Residual Lifetime
2. 07Regenerative Processes and Cycle-Based Analysis
3. 08The Inspection Paradox and Length-Biased Sampling
4. 09The Newsvendor Problem and Inventory Control
5. 10Exponential Tilting and Esscher Transforms
6. 11Heavy Traffic and Diffusion Approximations for Queues
7. 12Jackson Networks and Queueing Networks
8. 13Kingman's Formula for G/G/1 Queues
9. 14M/M/c Queues and the Erlang C Formula
10. 15Markov-Modulated Poisson Processes
11. 16Mixing Times and Convergence to Stationarity
12. 17Reliability Functions and System Redundancy
13. 18Ruin Probabilities With Heavy-Tailed Claims
14. 19Semi-Markov Processes
15. 20Subexponential Distributions and the Single Big Jump
16. 21The Branching Ratio and Critical Cascades
17. 22The Gittins Index
18. 23The House-Selling Problem and Reservation Values
19. 24The M/G/1 Queue and the Pollaczek-Khinchine Formula
20. 25The Prophet Inequality