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I'm applying for a job at a sports forecaster on the mathematical modeling side. My interview ended with the handing out of a test for which I have a week.

With only a MSc thesis on Sasaki-Einstein manifolds in relation to the AdS-CFT correspondence, my background is far from the interest of the casino. On related subjects, I have poor and basic knowledge of information theory, statistics, statistical mechanics and thermodynamics. Please, recommend to me books or other materials that will allow me to

  1. Understand time series, recognize standard processes defined by time series, and calculate their moments.

  2. Understand the biased and consistent estimators of a distribution defined up to a vector of unknown parameters and show that the mean of a random sample of the said distribution satisfies the conditions for both estimators.

  3. Undestand i.i.d. random variables and the method of maximum likelihood.

  4. Understand Martingales and Wiener processes together with examples, apply Ito's formula to solve SDE's, understand stable Levy processes, show that Weiner processes are Martingales.

During the interview we realized that I never encountered the Fokker-Plank equation nor the Kolmogorov-Smirnov test. In addition to a quick and dirty manual for the exam, I appreciate recommendations for in-depth materials exploring statistical models, statistical tests, the Fokker-Plank equation, Brownian motion and the relation between complex analysis and random variables, be such readings related to forecasting, pure mathematics or most curiously to physics.

Should I look at any of the books here, or in Brownian Motion - An Introduction to Stochastic Processes by René L. Schilling and Lothar (suggested [here])2?

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For Point $1$:

To be honest, Time Series is one of those nice topics that flows on from regression. Most books tend to be experimental analysis but I recommend Time Series Analysis which actually goes into the theory and best part is the simple worked examples in R. Such a great idea!

For Point $2$:

I recommend Statistical Inference Casella for all your Estimator needs. I wish it had a solution manual for free :(

For Point $3$:

This site will give you a good quick rundown plus some special (start with this): Random Services.

If you want a good rundown of everything, try Mathematical Statistics With Applications 7th Edition.

For Point $4$:

As a start, this is good for martingales and Ito's, BM etc. Financial Calculus an Introduction To Derivative Pricing.

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