Please enable JavaScript.
Coggle requires JavaScript to display documents.
Reading 08: Probability Concepts (Bayes Formula (\(P(I|O)=…
Reading 08: Probability Concepts
General terms
Random Variable
: an uncertain number.
Outcome
is a value 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.
Dependent vs Independent events
Independent events
: Knowing the outcomes of one tells you nothing about the probability of the other(
\(P\left ( A\mid B \right ) = P\left ( A \right )\)
Dependent event
: Knowing the outcomes of one tells you somethings about the probability of the other(
\(P\left ( A\mid B \right )\neq P\left ( A \right )\)
Probability
2 defining properties of probability
Sum the probabilities of all mutually exclusive and exhaustive events = 1
\(0\leq\) Probability of any event \(\leq 1\)
Distinguish based on input
Empirical
: based on analysis of data
Priori
: based on reasoning, not experience
Subjective
: based on personal guess.
Odds
Odd for X
= \(\frac{P(X)}{1-P(X)}\)
Odd against X
= \( \frac{1-P(X)}{P(X)}\)
Conditional vs Unconditional
Unconditional Probability
: probability of an event regardless of the outcomes of other events
Conditional Probability
P(A|B) is the probability of an event A occurring given that event B has occurred.
Probability rules
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.
Covariance of any variable with itself is its variance.
Sample Covariance
:
\( COV_{AB}=\frac{\sum_{t=1}^{n} \left | R_{t,A}-\bar{R_{A}} \right | \times \left | R_{t,B}-\bar{R_{B}} \right | }{n-1}\)
Covariance with Joint Probability Function
: \(Cov(X_{i}Y_{i})=\sum_{i=1}^{n}\left [ P(X_{i}Y_{i})\times [X_{i}-E(X)]\times [Y_{i}-E(Y)]\right ]\)
Correlation
Measures the linear relationship between 2 random variables
ranging from (-1) to (+1)
\(Corr(A,B)=\frac{Cov(A,B)}{\sigma _{a}\times\sigma _{b} }\)
Notation
\(\rho \) (rho) is population correlation coefficient
r
is the sample correlation coefficient
Scatter Plots
Shows relationship between 2 variables by plotting paired observations (one in x-axis and one in y-axis)
Interpreting Correlation
Correlation does not imply causation
Variable with strong
NONLINEAR
relationships may have correlation near zero
Spurious Correlation
Correlation may results from chance
Both variables may be related to a 3rd variable
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 return
\(E(R_{p})=W_{1}\times E(R_{1})+W_{2}\times E(R_{2})+...+W_{N}\times E(R_{N})\)
is a weighted average of the expected returns on the assets in the portfolio where the weights are proportions of portfolio values
Variance
\( Var(R_{p})=W_{1}^{2}\times \sigma_{1}^{2} + W_{2}^{2}\times \sigma_{2}^{2} + 2 \times W_{1}\times W_{2} \times Cov_{1,2} = W_{1}^{2}\times \sigma_{1}^{2} + W_{2}^{2}\times \sigma_{2}^{2} + 2 \times W_{1}\times W_{2} \times\sigma_{1}\times \sigma_{2}\times Corr_{1,2} \)
If 2 assets have negative Covariance (aka. negative correlation), the Portfolio Variance reduces
Covariance
\(Cov(R_{A},R_{B}) = \sum_{1}^{i}\sum_{1}^{j}P(R_{A,i},R_{B,j})\times (R_{A,i}-E(R_{A}))\times (R_{B,j}-E(R_{B}))\)
Bayes Formula
\(P(I|O)= \frac{P(O|I)}{P(O)}\times P(I)\)
We can update expectation of probability with new information
"What is the chance of I, given that O?"
P(O) = simple addition of P(O) in each scenario (as they are independent)
P(O|I) : the single probability of O given I (shown in the probability tree)
Counting Method
Factorial
No.ways to order N object
4! = 4 x 3 x 2 x 1
Factorial for labeling
Label A objects, B objects, C objects from a group of N
(N = A + B + C):
\(\frac{N!}{A! \times B! \times C!}\)
To pick r objects from a set of n
Permutation
: Order matters
\(nPr=\frac{n!}{(n-r)!\times r!}\)
Combination
: Order doesn't matter
\(nCr=\frac{n!}{(n-r)!}\)
