Latest revision as of 12:04, 22 November 2011

ECE302 Cheet Sheet number 1

You can get/put ideas for what should be on the cheat sheet here. DO NOT SIGN YOUR NAME

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]

Back to ECE302 Fall 2008 Prof. Sanghavi

@@ Line 1: / Line 1: @@
-You can get/put ideas for what should be on the cheat sheet here.
+[[Category:ECE302Fall2008_ProfSanghavi]]
+[[Category:probabilities]]
+[[Category:ECE302]]
+[[Category:cheat sheet]]
-=== Axioms of probability (finite spaces, infinite spaces) ===
+=[[ECE302]] Cheet Sheet number 1=
+You can get/put ideas for what should be on the cheat sheet here. <b> DO NOT SIGN YOUR NAME </b>
-<math> P(A) \geq 0 </math> for all events A
+'''Sample Space, Axioms of probability (finite spaces, infinite spaces)'''
-=== Sequential and continuous probability models ===
+. <math> P(A) \geq 0 </math> for all events A
-=== Properties of probability laws ===
+. <math>P(\omega)=1</math>
-=== Conditional probability ===
+. If A & B are disjoint then <math>P(A\cup B)=P(A)+P(B)</math>
-=== Independence ===
-=== Conditional Independence ===
+'''Properties of Probability laws'''
-=== Random Variables ===
-=== Probability mass functions ===
+'''Definition of conditional probability, and properties thereof'''
-=== Common random variables (Bernoulli, binomial, geometric) and how they come about ===
+<math>P(A|B) = \frac{P(A \cap B)}{P(B)}</math>
-=== Functions of random variables ===
+Properties:
-=== Mean and Variance, and their properties ===
+) <math>P(A|B) \ge 0</math>
-=== Joint PMFs of more than one random variable ===
+) <math>P( \Omega |B) = 1\!</math>
+) if A1 and A2 are disjoint <math>P(A1 \cup A2|B) = P(A1|B) + P(A2|B)</math>
+'''Bayes rule and total probability'''
+<math>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)</math>
+'''Definitions of Independence and Conditional independence'''
+Independence:
+A & B are independent if <math>P(A\cap B)=P(A)P(B)</math>
+side note: if A&B are independent then P(A|B)=P(A)
+Conditional Independence:
+A&B are conditionally independent given C if <math>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)}</math>
+'''Definition and basic concepts of random variables, PMFs'''
+Random Variable: a map/function from outcomes to real values
+Probability Mass Function (PMF)
+<math>P_X (x) = P(X=x)</math>
+'''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
+<math>P(X=k) = p(1-p)^{(k-1)}</math> for k>=1
+<math> E[X] = 1/p \!</math>
+<math>Var(x)=\frac{(1-p)}{p^2}</math>
+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
+<math>E[X]=\sum_X x P_X (x)</math>
+<math>E[ax+b]=aE[x]+b</math> where a and b are constants
+<math> Var(X) = E[X^2] - (E[X])^2 \!</math>
+<math>Var(ax+b)=a^2 Var(x) </math>
+'''Joint PMFs of more than one random variable'''
+Joint Probability Mass Function
+<math>P_{XY}(x,y)=P({X=x}\cap {Y=y})</math>
+PX(x)=(SUM of all y)[PXY(x,y)]
+PY(y)=(SUM of all x)[PXY(x,y)]
+<math>E[g(X,Y)]=\sum_{X,Y} g(X,Y)P_{XY}(x,y)</math>
+E[ax+by]=aE[x]+bE[y]
+----
+[[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]]

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Latest revision as of 12:04, 22 November 2011

ECE302 Cheet Sheet number 1

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