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Summary

Distribution
    Uniform Distribution (continuous)
    Notation
    \(Unif([a,b])\)
    		Here
    \(a,b\)
    are parameters
    Where
    \(-\infty < a < b < +\infty\)
    PDF
    \(\left\{\begin{matrix} \frac{1}{b-a}&; \text{for all x}\in [a,b] \\ 0& \text{otherwise} \end{matrix}\right. \)
    Mean
    \(\frac{a+b}{2}\)
    Median
    \(\frac{a+b}{2}\)
    Varience
    \(\frac{(b-a)^2}{12}\)
    Bernoulli Distribution
    Notation
    \(Ber(p)\)
    \(0<p<1\)
    PDF
    \(p^X(1-p)^{1-X}\)
    		Here
    \(X\)
    is the random variable
    Mean
    \(p\)
    Varience
    \(p(1-p)\)
    Binomial Distribution
    Notation
    \(Bin(n,p)\)
    \(n\in\{0,1,2,...\}\)
    ; number of trials
    \(p\in(0,1)\)
    ; it's probability of success for each trial
    PMF
    \(\begin{pmatrix} n\\ k \end{pmatrix} p^k(1-p)^{n-k}\)
    \(k\)
    is number of successes
    \(k\in\{0,1,...,n\}\)
    Mean
    \(np\)
    Varience
    \(np(1-p)\)
    Geometric Distribution
    Notation
    \(Geo(p)\)
    \(p\)
    is the success probability
    \(0 < p < 1\)
    PMF
    \((1-p)^{k-1}p\)
    \(k\)
    is number of failures
    \(k\in\{0,1,2,3,...\}\)
    Mean
    \(\frac{1}{p}\)
    Varience
    \(\frac{1-p}{p^2}\)
    Beta Distribution
    Notation
    \(Beta(\alpha,\beta)\)
    \(\alpha >0\ and\ \beta > 0\)
    PDF
    \(C\times x^{(\alpha-1)}(1-x)^{(\beta-1)}\mathbb{1}(x\in[0,1]) \)
    \(C\)
    is a constant
    Mean
    \(\frac{\alpha}{\alpha+\beta}\)
    Varience
    \(\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)
    Gaussian Distribution
    Notation
    \(\mathcal{N}(\mu,\sigma^2)\)
    		Here
    \(\mu,\sigma^2\)
    are parameters
    Where
    \(-\infty < \mu < +\infty\)
    and
    \(\sigma^2 \gt 0\)
    PDF
    \[f(x)=\frac{1}{\sigma \sqrt{2\pi }} \exp \left(-\frac{(x-\mu )^2}{2 \sigma ^2}\right)\]
    \(-\infty \lt x \lt \infty\)
    Mean
    \(\mu\)
    Varience
    \(\sigma^2\)
    Exponential Distribution
    Notation
    \(Exp(\lambda)\)
    		Here
    \(\lambda\)
    is parameters
    Where
    \(\lambda \gt 0\)
    PDF
    \[\lambda e^{-\lambda t}\]
    \(t \gt 0\)
    Mean
    \(\frac{1}{\lambda}\)
    Varience
    \(\frac{1}{\lambda^2}\)
Visit Chapter
Law of large numbers(LLN) Say we have
\(n\)
observations.
\(X,X_1,X_2,X_3,....,X_n\)
be I.I.D. random varibles, and
\(\mathbb{E}[X]=\mu\)
Then:
\[\overline{X}_n:=\frac{1}{n}\sum _{i=1}^ n X_ i \xrightarrow [n\to \infty ]{\mathbb{P},\text{ a.s.}} \mu\]
Visit Chapter
Central Limit Theorem(CLT) Say we have
\(n\)
observations
\(X,X_1,X_2,X_3,....,X_n\)
be
\(I.I.D.\)
random varibles,
\(\mathbb{E}[X]=\mu\)
and
\(Var(X)=\sigma^2\)

\[\sqrt{n} \frac{\overline{X}_n-\mu }{\sigma } \xrightarrow [n\to \infty ]{(d)} \mathcal{N}(0,1) \]
equivalently:
\[\sqrt{n} (\overline{X}_n-\mu ) \xrightarrow [n\to \infty ]{(d)} \mathcal{N}(0,\sigma^2) \]
And the quantity
\(\sigma^2\)
is called asymptotic variance of
\(\overline{X}_n\)
In progress ...