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Sample Space, Axioms of probability (finite spaces, infinite spaces)
1. $ P(A) \geq 0 $ for all events A
2. $ P(\omega)=1 $
3. If A & B are disjoint then $ P(A\cup B)=P(A)+P(B) $
Properties of Probability laws
Definition of conditional probability, and properties thereof
$ P(A|B) = \frac{P(A \cap B)}{P(B)} $
Properties:
1) $ P(A|B) \ge 0 $
2) $ P( \Omega |B) = 1\! $
3) if A1 and A2 are disjoint $ P(A1 \cup A2|B) = P(A1|B) + P(A2|B) $
Bayes rule and total probability
$ P(B)=P(B\cap A_1) + P(B \cap\ A_2) +...+P(B\cap A_n)= P(B|A_1)P(A)+P(B|A_2)P(A_2)+...+P(B|A_n)P(A_n) $
Definitions of Independence and Conditional independence
Independence: A & B are independent if $ P(A\cap B)=P(A)P(B) $ side note: if A&B are independent then P(A|B)=P(A)
Conditional Independence: A&B are conditionally independent given C if $ P(A\cap B|C)=P(A|C)P(B|C)=\frac{P(A\cap C)}{P(C)} \frac{P(B\cap C)}{P(C)} $
Definition and basic concepts of random variables, PMFs
Random Variable: a map/function from outcomes to real values
Probability Mass Function (PMF) let z be a script x $ P_x (z) = P(x=z) $
The common random variables: bernoulli, binomial, geometric, and how they come about in problems. Also their PMFs.
Geometric RV:
where X is # of trials until the first success
$ P(X=k) = p(1-p)^{(k-1)} $ for k>=1
$ E[X] = 1/p \! $
$ Var(x)=\frac{(1-p)}{p^2} $
Binomial R.V. "many biased coins"
with parameters n and p where n is the number of outcomes.
where X is # of successes in n trials and is the sum of independent, identically distributed outcomes.
P(X=k) = nCk * p^k * (1-p)^(n-k) for k=0,1,2,...n
E[X]=np VAR[X]=np(1-p)
Bernoulli R.V "one biased coin" with parameter p
X=1 if A occurs and X=0 otherwise
P(1)=1
E[x]=p
Var(X)=p(1-p)
Definition of expectation and variance and their properties let z be script x again
$ E[x]=\sum_x z P_x (z) $
$ E[ax+b]=aE[x]+b $ where a and b are constants
$ Var(X) = E[X^2] - (E[X])^2 \! $
$ Var(ax+b)=a^2 Var(x) $
Joint PMFs of more than one random variable
PX(x)=(SUM of all y)[PXY(x,y)]
PY(y)=(SUM of all x)[PXY(x,y)]
