Normal distribution or Gaussian
The normal distribution, Gauss distribution or Gaussian distribution, is discrete v.a. Z that measures the area included in the function represented by the Gaussian bell
Its probability function is:
\frac{1}{\sqrt{2 \cdot \pi}\cdot \sigma}\int^{+\infty}_Z e^m dx with m =-\frac{(x - \mu)^2}{2 \cdot \sigma^2}
E(\xi) = \mu
\sigma^2(\xi) = \sigma^2
\sigma(\xi) = \sigma
Properties of Normal
- Is symmetric with respect to the axis x = \mu, P(\sigma > \mu) = P(\sigma < \mu) = \frac{1}{2}
- When x \rightarrow \pm\infty we have an asymptote general y = 0
- It has points of inflection at x = \mu = \sigma
- Any v.a. built as a linear combination of normal v.a. also follows a normal distribution
Calculation a Normal
Normal typed
It \xi v.a. we will call typing another v.a. when:
Z = \frac{\xi - \mu \cdot \xi}{\sigma \cdot \xi}
If we type a z v.a. we have to:
If \xi \approx N(\xi, \sigma) \Rightarrow Z = \frac{\xi - \mu}{\sigma} \Rightarrow N(0, 1)
The probability function of the typed normal is:
P(Z > z) = \frac{1}{\sqrt{2 \cdot \pi}}\int^{+\infty}_Z e^m dx with m = -\frac{x^2}{2}, \forall x \in \mathbb{R}
E(\xi) = 0
\sigma^2(\xi) = 1
\sigma(\xi) = 1
Properties of the typed Normal
- Is symmetric with respect to the axis x = 0, P(\sigma > 0) = P(\sigma < 0) = \frac{1}{2}
- When x \rightarrow \pm\infty we have a horizontal asymptote
- It has points of inflection at x = \pm 1
Calculation of a typed Normal
Notes for Normal
It \xi_1, \cdots, \xi_n \approx v.to. (\mu_i, \sigma_i) with \xi = a_0 + a_1 \cdot \xi_1 + \cdots + a_n \cdot \xi_n, \forall i \in \{1, \cdots, n\}
- E[\xi] = a_0 + a_1 \cdot \mu_1 + \cdots + a_n \cdot \mu_n
- \sigma^2[\xi] = a_1^2 \cdot \sigma_1^2 + \cdots + a_n \cdot \sigma_n^2 + 2a_{1 2} \cdot Cov(\xi_1, \xi_2) + \cdots
- If the \xi_i are independent or only incorreladas \sigma^2[\xi] = a_1^2 \cdot \sigma_1^2 + \cdots + a_n \cdot \sigma_n^2
- If, in addition, \xi_i \approx N(\mu_i, \sigma_i), \forall i \in \{1, \cdots, n\} then:
\begin{cases} \xi \approx N(\mu, \sigma)\text{ with }\mu = a_0 + a_1 \cdot \mu_1 + \cdots + a_n \cdot \mu_n \\ \sigma^2 = a_1^2 \cdot \sigma_1^2 + \cdots + a_n \cdot \sigma_n^2 + 2a_{1 2} \cdot Cov(\xi_1, \xi_2) + \cdots \end{cases}
- If, in addition, \xi_i \approx N(\mu_i, \sigma_i), \forall i \in \{1, \cdots, n\} and independent then:
\begin{cases}\mu = a_0 + a_1 \cdot \mu_1 + \cdots + a_n \cdot \mu_n \\ \sigma^2 = a_1^2 \cdot \sigma_1^2 + \cdots + a_n \cdot \sigma_n^2 \end{cases}
It \xi_S = \xi_1 + \cdots + \xi_n
- E[\xi_S] = \mu_1 + \cdots + \mu_n
- \sigma^2[\xi_S] = \sigma_1^2 + \cdots + \sigma_n^2 + 2 \cdot Cov(\xi_1, \xi_2) + \cdots
- If the \xi_i are independent \sigma^2[\xi_S] = a_1^2 + \cdots + a_n^2
- If, in addition, \xi_i \approx N(\mu_i, \sigma_i), \forall i \in \{1, \cdots, n\} then:
\begin{cases} \xi_S \approx N(\mu, \sigma)\text{ with }\mu = \mu_1 + \cdots + \mu_n \\ \sigma^2 = a_1^2 \cdot \sigma_1^2 + \cdots + a_n \cdot \sigma_n^2 + 2a_{1 2} \cdot Cov(\xi_1, \xi_2) + \cdots \end{cases}
- If, in addition, \xi_i \approx N(\mu_i, \sigma_i), \forall i \in \{1, \cdots, n\} and independent then:
\begin{cases} \mu = \mu_1 + \cdots + \mu_n \\ \sigma^2 = \sigma_1^2 + \cdots + \sigma_n^2\end{cases}
It \xi_S v.a. independently and identically distributed with \xi_1, \cdots, \xi_n \approx v.a.i.i.d. (\mu, \sigma) and \xi_S = \xi_1 + \cdots + \xi_n \approx v.to.(n \cdot \mu, \sigma \cdot \sqrt{n})
- E[\xi_S] = \overbrace{\mu + \cdots + \mu}^{n\;\rm times} = n \cdot \mu
- \sigma^2[\xi_S] = \overbrace{\sigma^2 + \cdots + \sigma^2}^{n\;\rm times} = n \cdot \sigma^2
- If the \xi_i are v.a.i.i.d. and normal:
\xi_S \approx N(n \cdot \mu, \sqrt{n} \cdot \sigma)
Approaches
Approximation of the Binomial to the Normal
It B \approx B(n, p)
With B \approx number of successes in n equal and independent Bernoulli tests with probability of success p then:
B \approx B(n\cdot p, \sqrt{n \cdot p \cdot q})
Theorem of Moivre
It B \approx B(n, p) then:
\frac{B - m}{\sqrt{n \cdot p \cdot q}}\rightarrow N(0, 1)
So we have to:
E(B) = n \cdot p
\xi^2(B) = n \cdot p \cdot q
\xi(B) = +\sqrt{n \cdot p \cdot q}
It is considered a good approximation when n \cdot p \geq 5 and n \cdot q \geq 5 and then the Moivre Theorem is fulfilled with:
B \approx B(n \cdot p, \sqrt{n \cdot p \cdot q})
However, a discontinuity correction will have to be made to get the value sought, taking -0.5 if we look for the strictest or +0.5 in any other case
Examples:
P\{B < 4\} we will use P\{B < 3.5\}
P\{B \leq 4\} we will use P\{B \leq 4.5\}
P\{B > 4\} we will use P\{B > 4.5\}
P\{B \geq 4\} we will use P\{B \geq 4.5\}
The central limit theorem
\xi_1, \cdots, \xi_n \approx v.a.i.i.d. (\mu, \sigma) then:
\xi_T = \xi_1 + \cdots + \xi_n \approx v.to. (n \cdot \mu, \sqrt{n} \cdot \sigma) always happens
Theorem Lery-Lidenberg
It \{\xi_i\}, i \in N succession of v.a.i.i.d. then:
S_n = \xi_1 + \cdots + \xi_n \approx \frac{S_n - n \cdot \mu}{\sqrt{n} \cdot \sigma} \rightarrow N(0, 1)
It is considered a good approximation when n \geq 30 and then the Lery-Lidenberg Theorem is fulfilled by approx. normal any probability of the type:
\frac{S_n - n \cdot \mu}{\sqrt{n} \cdot \sigma}
Taylor series for Normal distribution
Approximation of Abramowitz and Stegun (1964) known as "Best Hastings Approach"
\tiny P(x) = 1 - \phi(x)(b_1 \cdot t + b_2 \cdot t^2 + b_3 \cdot t^3 + b_4 \cdot t^4 + b_5 \cdot t^5) + \epsilon(x)
\begin{cases} \phi(x) = \frac{1}{\sqrt{2 \cdot \pi}} \cdot e^{-\left(\frac{1}{2}\right) \cdot x^2} \\ t = \frac{1}{1 + (b_0 \cdot x)} \\ b_0 = 0.2316419 \\ b_1 = 0.319381530 \\ b_2 = -0.356563782 \\ b_3 = 1.781477937 \\ b_4 = -1.821255978 \\ b_5 = 1.330274429 \\ \|\epsilon(x)\| < 7.5 \cdot 10^{-8} \end{cases}
Replacing us we have:
\tiny P(x) = 1 - \left(\frac{1}{\sqrt{2 \cdot \pi}} \cdot e^{-\left(\frac{1}{2}\right) \cdot x^2}\right) \cdot \left(\left(\frac{0.319381530}{1 + 0.2316419 \cdot x}\right) + \left(\frac{-0.356563782}{1 + 0.2316419 \cdot x}\right)^2 + \left(\frac{1.781477937}{1 + 0.2316419 \cdot x}\right)^3 + \left(\frac{-1.821255978}{1 + 0.2316419 \cdot x}\right)^4 + \left(\frac{1.330274429}{1 + 0.2316419 \cdot x}\right)^5 \right) + \epsilon(x)