Graduationwoot

Dragon Notes

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  UNDER CONSTRUCTION

Randomness & Probability:
Caveats





[Fair but dependent]

: Two fair coin tosses are not necessarily independent
Consider a fair toss experiment with a penny and a nickel, with tails mapped to 0 and heads to 1. Assume the joint PMF as below:\(\vplup\)
\(j=0\hspace{35px} j=1\)\(p_X[i]\)
\(\begin{matrix}i=0\\i=1\end{matrix}\)\(\begin{matrix}\frac{3}{8}\\ \frac{1}{8}\end{matrix}\hspace{50px}\) \(\begin{matrix}\frac{1}{8}\\ \frac{3}{8}\end{matrix}\)\(\begin{matrix}\frac{1}{2}\\ \frac{1}{2}\end{matrix}\)
\(p_Y[j]\)\(\frac{1}{2}\hspace{60px} \frac{1}{2}\)

Examining the PMF confirms the coins are fair since \(p=1/2\) - however, as \(p_{X,Y}[0,0]=3/8\neq (1/2)(1/2)=p_X[0]p_Y[0]\), \(X\) and \(Y\) are dependent.

[Zero covar \(\neq\) indep]

: Independence implies zero covariance but zero covariance does not imply independence

\(E[g(X)]\neq g(E[X])\)


If \(\vplup g(X)=X^2\), then \(E[g(X)]=E[X^2]=6\) but \(g(E[X])=(E[X])^2=4\neq E[g(X)]\).

[Not all PMFs have expected values]


Consider the PMF \(\ds p_X[k]=\frac{4/\pi^2}{k^2},\ \ k=1,2,...\) Attempting to find the expected value produces
\(\ds E[X]=\frac{4}{\pi^2}\sum_{k=1}^{\infty}\frac{1\vphantom{1^A}}{k}\rightarrow \infty\)

[Assessing independence - mind PDF domain]


The joint PDF given by \(\vplup\)
\(\ds p_{X,Y}(x,y)=\left\{ \begin{array}{c} \begin{align} & 2\t{exp}[-(x+y)], && x\geq 0,\ y\geq 0,\t{ and } y < x \\ & 0, && \t{otherwise} \end{align} \end{array} \right.\)

is not factorable; the region in the \(xy\)-plane where \(p_{X,Y}(x,y)\neq 0\) cannot be written as \(g(x)h(y)\).

[Covariance vs. correlation]


Covariance \(=\) the expected product of deviations of two rv's from their individual expected values
Correlataion \(=\) the strength of linear association between two rv's
  - Measures how much variables change together
  - Has dimensions; \([\t{cov}] = [\t{var}_1][\t{var}_2]\)
  - Positive covariance \(\hspace{5px}=\) \(\uparrow (\t{var}_1)\ \rightarrow\ \uparrow (\t{var}_2)\)
  - Negative covariance \(=\) \(\uparrow (\t{var}_1)\ \rightarrow\ \downarrow (\t{var}_2)\)
  - Independent rv's \(\rightarrow \t{cov}=0\)
  - Is dimensionless
  - Independent rv's \(\not\rightarrow \t{corr}=0\)

[Uncorr & indep]

Uncorrelated implies independence only for multivariate Gaussian PDF, even if marginal PDF's are Gaussian

[Stationarity & realizations]

It is impossible to determine if a random process is stationary from a single realization
\(\vplup\)We cannot determine if a coin is fair by observing that one of the tosses was a head. Multiple realizations of the coin tossing experiment are required - so is with random processes.





Dragon Notes,   Est. 2018     About

By OverLordGoldDragon