Introduction

Random variables, sure variables

stochastic process → time evolution of random variables X(t)
deterministic process → time evolution of sure variables. X (t)

Assigning probability →

• statistical or frequency
• inductive-equal a priori

independent event.

expected values

moments

$⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=\underset{i}{\sum \; <!--\; n-ary\; summation\; -->}{x}_{i}P\left(x_i\right)$

Probabilities → mean (sure variables)

Mean(first moment), Variance (2nd moment); skewness (3rd moment); kurtosis (4th moment)

$var\left(X\right)=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->{\left(X-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\right)}^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^2={\mathrm{\sigma \; <!--\; greek\; small\; letter\; sigma\; -->}}^2$

$\begin{array}{}cov\left(X,Y\right)=\left(X-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\right)\times \; <!--\; multiplication\; sign\; -->\left(Y-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->Y⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\right)\\ cov=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->XY⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->Y⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\\ cor\left\{x,y\right\}=\frac{cov\left(x,y\right)}{\sqrt{var\left(x\right)var\left(y\right)}}\\ -1<cor\left\{x,y\right\}<1\end{array}$

Covariance variance of 2 different variable.

Correlation coefficient shows similarity of 2

random variable

Variance sum theorem

$\begin{array}{}var\left\{X,Y\right\}=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->{\left(\left(X+Y\right)-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X+Y⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\right)}^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\\ var\left\{X,Y\right\}=var\left\{X\right\}+var\left\{Y\right\}+cov\left\{X,Y\right\}\end{array}$

How do we combine different meajurements of the same random quantity?

$\begin{array}{}⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_3⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\\ var\left(X_3\right)<var\left(X_1\right)\\ var\left(X_3\right)<var\left(X_2\right)\\ \stackrel{-}{x}=\frac{1}{3}\times \; <!--\; multiplication\; sign\; -->\left(x_1+x_2+x_3\right)\end{array}$

But lets assume variance of one of them less than others. If we take average do we des troy the best mea jurement?

The issue is resolved realizing average is the average of random variable

Now, average

$\begin{array}{}⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->\stackrel{-}{X}⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X_3⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\\ var\left\{\stackrel{-}{X}\right\}=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->{\left(\stackrel{-}{X}\right)}^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->\stackrel{-}{X}{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^2\\ \begin{array}{}var\left\{\stackrel{-}{X}\right\}=\frac{1}{9}\times \; <!--\; multiplication\; sign\; -->\left[var\left\{X_l\right\}+?\right]\\ \left[var\left\{X_2\right\}+var\left\{X_3\right\}\right]\end{array}\end{array}$

$\stackrel{-}{X}=\frac{1}{3}\times \; <!--\; multiplication\; sign\; -->\left(X_1+X_2+X_3\right)$

If variance of sum is less than each variable than averaging gives more precise measurement

But notice that if variance of each meajurement is same than averaging always gives more precise result.

problems

1. An unbiased dice is given. find mean, variance standards deviation, skewness an kurtosis of X

$\begin{array}{}⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=\underset{i=1}{\overset{6}{\sum \; <!--\; n-ary\; summation\; -->}}{x}_{i}p\left(x_i\right)=\frac{\left(1+2+3+4+5+6\right)}{6}=3.5\\ ⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=\frac{1+2^2+3^2+4^2+5^2+6^2}{6}=15.167\\ var\left(X\right)=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^2=15.167-{\left(3.5\right)}^2=2.917\\ std=\sqrt{2.917}=1.708\\ ⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^3⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=\frac{1+2^3+3^3+4^3+5^3+6^3}{6}=73.5\\ skewness=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^3⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^3=73.5-{\left(3.5\right)}^3=30.625\\ ⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^4⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=\frac{1+2^4+3^4+4^4+5^4+6^4}{6}=379.167\\ kurtosis=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^4⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^4=379.167-{\left(3.5\right)}^4=229.105\end{array}$

Two random variables X and Y are related by Y = m X+b

1. his means every realizationxi of X is related to realization yi of Y. Prove that

$cor\left\{x,y\right\}=\frac{m}{\sqrt{m^2}}=sign\left(m\right)$

$\begin{array}{}cor\left\{x,y\right\}=\frac{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->XY⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->Y⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}{\sqrt{var\left(X\right)var\left(Y\right)}}=\frac{m}{\sqrt{m^2}}=sign\left(m\right)\\ var\left(X\right)=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\\ var⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->Y⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->=⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->{\left(mX+b\right)}^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->-⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->mx+b{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^2\\ var\left(Y\right)=m^2⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X^2⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->+2mb⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->+b^2-\left(m^2⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X{⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}^2+2mb⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->X⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->+b^2\right)\\ var\left(Y\right)=m^{2}var\left(X\right)\end{array}$