- $\mu =E\left ( X \right )$
- $\sum P\left ( x \right )=1$ where
-
- 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)
$=E\left ( X^{2} \right )-\mu ^{2}$
$=\frac{\left ( b-a \right )^{2}}{12}$. [for rectangular distribution]
- $Var\left ( X_{N} \right )=\frac{1}{N}\sum_{i=1}^{N}\left ( X_{I}-\mu \right )^{2}$ where $\mu =\frac{X_{1}+X_{2}+...X_{N}}{N}$
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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
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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