cli-esej.md

1. Pomembne neenakosti

1. Cauchy-Schwarz inequality

$$E(XY)\leq \sqrt{E(X^2)E(Y^2)}$$

• v primeru neodivisnosti spremenljivk velja enakost

Dokaz

• pride direktno iz lastnosti korelacijskega koeficienta primer $E(Y)=E(X)=\mu = 0$ $$|r(X,Y)|=\left| \frac{E((X-E(X))(Y-E(Y)))}{\sqrt{E(X^2)-E^2(X)}\cdot \sqrt{E(Y^2)-E^2(Y)}} \right| =\left| \frac{E(XY)}{\sqrt{E(X^2)E(Y^2)}} \right| \leq 1$$

2. Jensenova neenakost

Ce je $g$ konveksna ($g''(x)>0$), potem $$E(g(X))\geq g(E(X))$$ Ce je $g$ konkavna potem: $$E(g(X))\leq g(E(X))$$

Dokaz

$$g(x) \geq a+bx, \text{ (Vsaka tangenta je pod krivuljo (konveksnost))}$$ $$g(X) \geq a+bX$$ $$E(g(X))\geq E(a+bX)=a+bE(X)=a+b\mu=g(\mu)=g(E(X))$$

Primeri

$$E(X^2) \geq (E(X))^2$$ $$E\left(\frac{1}{X}\right) \geq \frac{1}{E(X)}$$ $$E(\log(X))\leq \log(E(X))$$

3. Markova neenakost

$$P(|X|\geq a) \leq \frac{E(X)}{a}, \forall a >0$$

Dokaz

$$a I_{|X|\geq a} \leq |X|, \text{vedno drzi}$$

• I = 1 ce $|X|>a$
• I = 0 ce $|X|<a$ $$aE(I_{|X|>a}) \leq E(X)$$

4. Chebyshev inequality

$$P(|X-\mu| > a) \leq \frac{\text{Var}(X)}{a^2}$$ $$P(|X-\mu|>c\cdot\text{SD}(X))\leq \frac{1}{c^2}$$

2. Law Of Large Numbers - Bernoullijev zakon velikih stevil

Naj bojo $X_1, X_2, \dots$ enake slucajne spremenljivke z pricakovano vrednostjo $\mu$ in varianco $\sigma^2$.

Naj bo $\overline{X}=\frac{1}{n} \sum\limits_{i=1}^{n} X_i$ vzorcno povprecje

(Mocen) Zakon velikih stevil

$$\overline{X} \rightarrow \mu, \text{ ko } n \rightarrow \infty \text{ z verjetnostjo 1}$$

(Sibek) Zakon velikih stevil

$$\forall c>0, P(|\overline{X}-\mu| > c) \rightarrow 0$$

Dokaz

Uporabimo Chebyshevo neenakost $$P(|\overline{X}_ n-\mu| > c) \leq \frac{Var(\overline{X})}{c^2}=\frac{\frac{1}{n^2} n\sigma^2}{c^2}=\frac{\sigma^2}{nc^2}\rightarrow 0$$

(Primer)

$$X_i \sim B(p)$$ $$\overline{X} = \sum\limits_{i=1}^n X_i = p, \text{ za n } \rightarrow \infty$$

3. CLI