ECE302 Cheet Sheet number 1

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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) $P_X (x) = P(X=x)$

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)=p

E[x]=p

Var(X)=p(1-p)

Definition of expectation and variance and their properties

$E[X]=\sum_X x P_X (x)$

$E[ax+b]=aE[x]+b$ 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

Joint Probability Mass Function

$P_{XY}(x,y)=P({X=x}\cap {Y=y})$

PX(x)=(SUM of all y)[PXY(x,y)]

PY(y)=(SUM of all x)[PXY(x,y)]

$E[g(X,Y)]=\sum_{X,Y} g(X,Y)P_{XY}(x,y)$

E[ax+by]=aE[x]+bE[y]