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Random Variable

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  • $\mu =E\left ( X \right )$
  • $\sum P\left ( x \right )=1$ where
  • $x$ 1 2 …...
    $P(x)$      

     

  • Standard deviation=    $\sigma$     =$\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}\left ( x_{i}-\bar{x} \right )^{2}}$ [for large data like population , it will be                                                                                           $\left ( N-1 \right )$, otherwise it will be $N$]
  • varience $Var\left ( X \right )=\sigma ^{2}$

 

Continuous Random Variable

  • $E\left ( X \right )=\int_{a}^{b}xf\left ( x \right )dx$
  • Probability  or   PDF  $P\left ( X> x \right )=1-P\left ( X\leq x \right )$  where $P\left ( X\leq x \right )\int_{-\infty}^{x}f\left ( y \right )dy$

 

 

Discrete Random Variable

  • $E\left ( X \right )=\sum_{i=1}^{n}x_{i}P_{i}$

 

Properties of $E\left ( X \right )$

[All are same for discrete and continuous random variable] 

  • $E\left ( X+Y \right )=E\left ( X \right )+E\left ( Y \right )$
  • $E\left ( aX+b \right )=aE\left ( X \right )+b$

Instead of linearity of $E\left ( X \right )$ if $X=N\left ( \mu ,\sigma ^{2} \right )$ is a normal variable, we can standardize it $Z=\frac{X-\mu }{\sigma }$

  • $E\left ( X \right )=E\left ( \sigma Z+\mu \right )$ 

                                      $=\sigma E\left ( Z \right )+\mu$

                                       $=\mu$.          ....................................................................................................................(i)

 

Properties of Varience 

(All are Same for Discrete and Continuous Random Variable)

  • $Var\left ( X+Y \right )=Var\left ( X \right )+Var\left ( Y \right )$

  • $Var\left ( aX+b \right )=a^{2}Var\left ( X \right )$
  • $Var\left ( X \right )=E\left ( X^{2} \right )-E\left ( X \right )^{2}$.      where $E\left [ X^{2} \right ]=x^{2}.P\left ( X \right )$

                              $=E\left ( X^{2} \right )-\mu ^{2}$ 

                              $=\frac{\left ( b-a \right )^{2}}{12}$.  [for rectangular distribution]

  • $Var\left ( X_{N} \right )=\frac{1}{N}\sum_{i=1}^{N}\left ( X_{I}-\mu \right )^{2}$ where $\mu =\frac{X_{1}+X_{2}+...X_{N}}{N}$

--------------------------------------------------------------------------------------------------------------------------

Now, for Continuous Random Variable

  • $E\left ( X^{2} \right )=\int_{0}^{\alpha }x^{2}\lambda e^{-\lambda x}dx=\frac{2}{\lambda ^{2}}$                   ..........................................(ii)
  • $Var\left ( X \right )=E\left ( \left ( X-\mu \right )^{2} \right )$  or $E\left ( X-E\left ( X \right ) \right )^{2}$        -------------------------------------------(iii)
  • An example here

------------------------------------------------------------------------------------------------------------------------------

Uniform Random Variable

  • $E\left ( X \right )=\int_{a}^{b}xdx$ 
  • $f\left ( x \right )=\frac{1}{b-a}$
  • Varience  $=\frac{\left ( b-a \right )^{2}}{12}$.

Exponential Random Variable 

  • $E\left ( X \right )=\int_{a}^{b}\lambda e^{-\lambda .x }dx$ where pdf $f\left ( x \right )=\lambda e^{-\lambda x}$
  • If $X=[0,\alpha )$ $E\left ( X \right )=\frac{1}{\lambda }$

 

 

 

Normal Distribution

  • pdf $\phi \left ( z \right )=\frac{1}{\sqrt{2\pi }}e^{-\frac{z^{2}}{2}}$
  • $E \left ( z \right )=\int_{-\alpha }^{\alpha }\frac{1}{\sqrt{2\pi }}ze^{-\frac{z^{2}}{2}}dz=0$ where $N\left ( 0,1 \right )$ 

 

 

Binomial Distribution

  • mean $=np$
  • variance $=npq=\sigma^{2}$

 

Probability Density Function

  • $\mu =\int xf\left ( x \right ) dx$
  • $\int_{-\infty}^{\infty}f\left ( x \right ) dx=1$

 

 

Poisson Distribution

  • mean=variance [when $n$ is large and $p$ small]
  • $E\left [ X \right ]$ by derivative here

I have collected all formulas of random variable

Is all are correct and important ? Is anything missing plz tell me??

Can someone also derive (i),(ii),(iii) points, I am not getting proper reason behind them

Plz check it

 

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