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Reading 09: Probability Concepts (Probability (Distinguish based on input,…
Reading 09: Probability Concepts
General terms
Random Variable
: an uncertain value determined by chance.
Outcome
is an observed valued of random variable
Event
: set of 1 or more outcomes.
Mutually exclusive events
: 2 event cannot both occur P(AB)=0
Exhaustive events
: set of events that include all possible outcome.
Independent events
: 2 (or more) event whose probability of occurrence are not affected by others'
P(A|B)=P(A) and P(B|A)=P(B)
Probability
2 defining properties of probability
Sum the probabilities of all mutually exclusive events = 1
\(0\leq\) Probability of any event \(\leq 1\)
Distinguish based on input
Empirical
: Measure probability from observations or experiments
Priori
: measures probability from well-defined inputs.
Subjective
: informed guess.
Odds
if P(X)= A out of B trials
Odd for X
= \(\frac{P(X)}{1-P(X)}\)
Odd against X
= \( \frac{1-P(X)}{P(X)}\)
Conditional vs Unconditional
Unconditional Probability
: measures the chance of an occurrence ignoring any knowledge gained from previous or external events.
Conditional Probability
P(A|B) is the probability of an event A occurring given that event B has occurred.
Multiplication, addition, and total probability rule
Multiplication rule
: \(P(AB)=P(A|B)\times P(B)\)
Addition rule
: P(A or B) = P(A) + P(B) - P(AB)
Total probability rule
: \(P(A)=\sum_{i=1}^{N}\left [P(A|B_{i})\times P(B_{i}) \right ]\)
Covariance & Correlation
Covariance
: Measure how 2 random variables move together. It is the expected value of the product of the deviations of 2 random variables from their respective expected value.
\(Cov(A,B)=\sum_{i=1}^{N}\left [ P_{i}\times \left ( A_{i}-\overline{A} \right ) \times \left ( B_{i}-\overline{B} \right ) \right ]\)
Correlation
: Measure the association between 2 random variables; ranging from (-1) to (+1)
\(Corr(A,B)=\frac{Cov(A,B)}{\sigma _{a}\times\sigma _{b} }\)
Random variables & Portfolio
Random variable
Expected value
: \(E(X)=\sum_{i=1}^{n}[P_{i}(x_{i})\times X_{i}]\)
Variance
: \(\sigma_{x}^{2}=\sum_{i=1}^{n}\left [ P\left ( x_{i} \right )\times \left [ X_{i}-E(X) \right ]^{2} \right ]\)
Portfolio
Expected value
: \(E(R_{p})=W_{1}\times E(R_{1})+W_{2}\times E(R_{2})\)
Variance
: \( Var(R_{p})=W_{1}^{2}\times \sigma_{1}^{2} + W_{2}^{2}\times \sigma_{2}^{2} + 2 \times W_{1}\times W_{2} \times Cov_{1,2} = W_{1}^{2}\times \sigma_{1}^{2} + W_{2}^{2}\times \sigma_{2}^{2} + 2 \times W_{1}\times W_{2} \times\sigma_{1}\times \sigma_{2}\times Corr_{1,2} \)
Covariance with Joint Probability Function
: \(Cov(X_{i}Y_{i})=\sum_{i=1}^{n}\left [ P(X_{i}Y_{i})\times [X_{i}-E(X)]\times [Y_{i}-E(Y)]\right ]\)
Bayes Formula
: \(P(I|O)= \frac{P(O|I)}{P(O)}\times P(I)\)
Counting Method
Factorial
: No.ways to order N object
To pick a subset from a set
Permutation
: Order matters
\(nPr=\frac{n!}{(n-r)!\times r!}\)
Combination
: Order doesn't matter
\(nCr=\frac{n!}{(n-r)!}\)