Review of probability

Admistrivia

• Class web page:
  • Crude lecture notes. (No math notation though)
  • Homeworks once per week
  • schedule.
  • Readings (I won't announce them in class since I forget).
• If you haven't seen probability in a while, read chapters 1 and 2 carefully. Today's "review" will be kinda fast if it is new.
• I expect to spend 10 minutes a day on HW problems IF you ask me questions--otherwise I'll just ignore the homework.

Intro: What is a stochastic process anyway?

• Population peramid
• Example: google page rank and random web surfing
  • Pick a page at random. Call it X₀. (Note: even this step is hard. Mathematically it requires numbering all the web pages.)
  • Follow a link at random. This is X₁.
  • Follow a link at random. This is X₂.
  • Repeat.
  • Typical question: If X₀ = www.gosset.upenn.edu, what is the chance that X₁ = www.library.upenn.edu?
  • Typical question: If X₀ = www.gosset.upenn.edu, what is the chance that X₂ = www.library.upenn.edu?
  • What is the distribution of X₁? X₂? X_infinity? (Called your page rank on google)
  • Let A = set of all pornography. P(X₄ in A|X₀ in wharton)?

Technical definition of stochastic process

• What is a random variable? (map from sample space to real line)
• Definition is unimportant for only one RV, but if you have many of them the definition matters.
• Example: X₁,X₂,...,X_T.
• What is a stochastic process?
  • Just a indexed set of random variables!
  • Typically the index is time
  • examples: stock market, store purchase behavior, physics, chemistry.

Chapter 1: Review of probability

• Recall expectation: E(g(X)).
• Poisson distribution: Start with Taylor series for e^lambda.
  • Note: expecation is a sum
• Exponential distribution: Same parameter as poisson?!?
  • Note: expecation is now an integral

More details of conditional expectation and probability

• P(A|B) basic definition
• Simple definition of conditional expectation:
  • E(X|Y=y) = E(XI_Y=y)/E(I_Y=y)
  • Deconstruct it
  • E(X|Y=y) = E(XI_Y=y)/P(y)
• Problem II.E.1.5 puts all this together

Conditional expectation as a random variable

• Conditional expectation as a random variable (hard)
  • Z = e(Y) for some function e()--called measurability
  • E(ZI_Y=y) = E(XI_Y=y)
  • Call such an e(Y) E(X|Y)
• Conditional expectation as a random variable (easy)
  • define e(y) = E(X|Y=y)
  • E(X|Y) = e(Y)
• Smoothing lemma: E(X) = E(E(X)|Y))

Simple martingale

• E(X), E(X|Y), X
• X = bill back in white house, Y = Hillary wins SC