Till Normal Distribution

TODO : Standard normal distribution funtion Z

main.org

@@ -1,5 +1,6 @@
 #+TITLE: Probability and Statistics ( BTech CSE )
 #+AUTHOR: Anmol Nawani
+#+LATEX_HEADER: \usepackage{amsmath}
 
 # *Statistics
 
@@ -307,3 +308,271 @@ And we know P(A) = 0.5, P(B) = 0.3 and P(C) = 0.2. Also P(P|A) = 0.20, P(P|B) =
 + Some other Identity
 \[ P(\overline{A} \cap B) + P(A \cap B) = P(B) \]
 \[ P(A \cap \overline{B}) + P(A \cap B) = P(A) \]
+
+* Probability Function
+
+It is a mathematical function that gives probability of occurance of different possible outcomes. We use variables to represent these possible outcomes called *random variables*. These are represented by capital letters. Example, $X$, $Y$, etc. We use these random variables as:
+\\
+ Suppose X is flipping two coins.
+\[ X = \{HH, HT, TT, TH\}  \]
+We can represent it as,
+\[ X = \{0, 1, 2, 3\} \]
+
+Now we can write a probability function $P(X=x)$ for flipping two coins as :
+
+#+attr_latex: :align |c|c|c|
+|-----+----------|
+| $x$ | $P(X=x)$ |
+|-----+----------|
+|   0 |     0.25 |
+|   1 |     0.25 |
+|   2 |     0.25 |
+|   3 |     0.25 |
+|-----+----------|
+
+Another example is throwing two dice and our random variable $X$ is sum of those two dice.
+
+#+attr_latex: :align |c|c|c|
+|-----+----------------|
+| $x$ | $P(X=x)$       |
+|-----+----------------|
+|   2 | $1/36$         |
+|   3 | $2/36$         |
+|   4 | $3/36$         |
+|   5 | $4/36$         |
+|   6 | $5/36$         |
+|   7 | $6/36$         |
+|   8 | $5/36$         |
+|   9 | $4/36$         |
+|  10 | $3/36$         |
+|  11 | $2/36$         |
+|  12 | $1/36$         |
+|-----+----------------|
+
+
+** Types of probability functions (Continious and Discrete random variables)
+
+Based on the range of the Random variables, probability function has two different names.
+
++ For discrete random variables it is called Probability Distribution function.
++ For continious random variables it is called Probability Density function.
+
+* Proability Mass Function
+
+If we can get a function such that,
+
+\[ f(x) = P(X=x) \]
+
+then $f(x)$ is called a *Probability Mass Function* (PMF).
+
+** Properties of Probability Mass Function
+
+Suppose a PMF
+
+\[ f(x) = P(X=x) \]
+
+Then, 
+
+*** For discrete variables
+
+\[ \Sigma f(x) = 1 \]
+\[ E(X^n) = \Sigma x^n f(x) \]
+
+For $E(X)$, the summation is over all possible values of x.
+
+\[ Mean = E(X) = \Sigma x f(x) \]
+\[ Variance = E(X^2) - (E(X))^2 = \Sigma x^2 f(x) - ( \Sigma x f(x) )^2 \]
+
+To get probabilities
+
+\[ P(a \le X \le b) =   \sum_{a}^{b} f(x) \]
+\[ P(a < X \le b) =   (\sum_{a}^{b} f(x))  - f(a) \]
+\[ P(a \le X < b) =   (\sum_{a}^{b} f(x)) - f(b) \]
+
+Basically, we just add all $f(x)$ values from range of samples we need.
+
+*** For continious variables
+
+\[ \int_{-\infty}^{\infty} f(x) dx = 1  \]
+\[ E(X^n) = \int_{-\infty}^{\infty} x^n f(x) dx \]
+
+We only consider integral from the possible values of x. Else we assume 0.
+
+\[ Mean = E(X) = \int_{-\infty}^{\infty} x f(x) dx \]
+\[ Variance = E(X^2) - (E(X))^2 = \int_{-\infty}^{\infty} x^2 f(x) dx - ( \int_{-\infty}^{\infty} x f(x) dx )^2 \]
+
+To get probability from a to b (inclusive and exclusive doesn't matter in continious).
+
+\[ P(a < X < b) = \int_{a}^{b} f(x) dx \]
+
+** Some properties of mean and variance 
+
++ Mean
+\[ E(aX) = aE(X) \]
+\[ E(a) = a \]
+\[ E(X + Y) = E(X) + E(Y) ]
+
++ Variance
+If
+\[ V(X) = E(X^2) - (E(X))^2 \]
+Then
+\[ V(aX) = a^2 V(X) \]
+\[ V(a) = 0 \]
+
+* Moment Generating Function
+
+The moment generating function is given by
+
+\[ M(t) = E(e^{tX}) \]
+
+** For discrete
+\[ M(t) = \sum_{0}^{\infty} e^{tx} f(x) \]
+
+** For continious
+\[ M(t) = \int_{-\infty}^{\infty} e^{tx} f(x) dx \]
+
+** Calculations of Moments (E(X)) using MGF
+
+\[ E(X^n) = (\frac{d^n}{dt^n} M(t))_{t=0} \]
+
+
+* Binomial Distribution
+The use of a binomial distribution is to calculate a known probability repeated n number of times, i.e, doing *n* number of trials.
+A binomial distribution deals with discrete random variables.
+
+\[ X = \{ 0,1,2, .... n \} \]
+
+where *n* is the number of trials.
+
+\[ P(X=x) = \ ^nC_x\ (p)^x(q)^{n-x}  \]
+
+Here
+\[ n \rightarrow number\ of\ trials \]
+\[ x \rightarrow number\ of\ successes \]
+\[ p \rightarrow probability\ of\ success \]
+\[ q \rightarrow probability\ of\ failure \]
+\[ p = 1 - q \]
+
++ Mean
+\[ Mean = np \]
++ Variance
+\[ Variance = npq \]
++ Moment Generating Function
+\[ M(t) = (q + pe^t)^n \]
+
+** Additive Property of Binomial Distribution
+
+For an independent variable $X$. The binomial distribution is represented as
+
+\[ X ~ B(n,p) \]
+Here,
+\[ n \rightarrow number\ of\ trials \]
+\[ p \rightarrow probability\ of\ success \]
+
++ Property
+If given,
+\[ X_1 \sim  B(n_1, p) \]
+\[ X_2 \sim  B(n_2, p) \]
+Then,
+\[ X_1 + X_2 \sim B(n_1 + n_2, p) \]
+
++ *NOTE*
+If
+\[ X_1 \sim  B(n_1, p_1) \]
+\[ X_2 \sim  B(n_2, p_2) \]
+Then $X_1 + X_2$ is not a binomial distribution.
+
+** Using a binomial distribution
+We can use binomial distribution to easily calculate probability of multiple trials, if probability of one trial is known. Example, the probability of a duplet (both dice have same number) when two dice are thrown is $\frac{6}{36}$. \\
+Suppose now we want to know the probability of a 3 duplets if a pair of dice is thrown 5 times. So in this case :
+
+\[ number\ of\ trials\ (n) = 5 \]
+\[ number\ of\ duplets\ we\ want\ probability\ for\ (x) = 3 \]
+\[ probability\ of\ duplet\ (p) = \frac{6}{36} \]
+\[ q = 1 - p = 1 - \frac{6}{36} \]
+
+So using binomial distribution,
+\[ P(probability\ of\ 3\ duplets) = P(X=3) = \ ^5C_3 \left(\frac{6}{36}\right)^3 \left(\frac{30}{36}\right)^{5-3} \]
+
+* Poisson Distribution
+A case of the binomial distribution where *n* is indefinitely large and *p* is very small and *$\lambda = np$* is finite.
+
+\[ P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!}\ if\ x = 0, 1, 2 ..... \]
+\[ P(X=x) = 0\ otherwise \]
+
+\[ \lambda = np \]
+
++ Mean
+\[ Mean = \lambda \]
++ Variance
+\[ Variance = \lambda \]
++ Moment Generating Funtion
+\[ M(t) = e^{\lambda\left(e^{t}-1\right)} \]
+
+** Additive property
+If X_1, X_2, X_3..X_n follow poisson distribution with \lambda_1, \lambda_2, \lambda_3....\lambda_n \\
+Then, 
+\[ X_1 + X_2 + X_3...+X_n \sim \lambda_1 + \lambda_2 + \lambda_3 + ...+ \lambda_n \]
+
+* Exponential Distribution
+A continuous random distribution which has probability mass function
+
+\[ f(x) = \lambda e^{-\lambda x}\ ,\ when\ x \ge 0 \]
+\[ f(x) = 0 \ ,\ otherwise \]
+
+\[ where\ \lambda > 0 \]
+
++ Mean
+\[ Mean = \frac{1}{\lambda} \]
++ Variance
+\[ Variance = \frac{1}{\lambda^2} \]
++ Moment Generating Function
+\[ M(t) = \frac{\lambda}{\lambda - t} \]
+
+** Memory Less Property
+
+\[ P[X > (s + t) \mid X > t] = P(X > s) \]
+
+* Normal Distribution
+Suppose for a probability funtion with random variable X, having mean \mu and variance \sigma^2.
+We denote normal distribution using $X \sim N(\mu,\sigma)$ \\
+The probability mass funtion is 
+
+\[ f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right) \]
+
+\[ -\infty < x < \infty \]
+\[ -\infty < \mu < \infty \]
+\[ \sigma > 0 \]
+Here, $exp(x) = e^x$
+
++ Moment Generating Funtion
+\[ M(t) = exp\left(  \mu t + \frac{\sigma^2 t^2}{2}  \right) \]
+
+** Odd Moments
+
+\[ E(X^{2n + 1}) = 0 \ , \ n = 0, 1, 2, ... \]
+
+** Even Moments
+
+\[ E(X^{2n}) = 1.3.5....(2n-3)(2n-1) \sigma^{2n} \ , \ n = 0, 1, 2, ... \]
+
+** Properties
+
++ In a normal distribution
+\[ Mean = Mode = Median \]
+
++ For normal distribution, mean deviation about mean is
+
+\[ \sigma \sqrt{ \frac{2}{\pi}  } \]
+
+** Additive property
+Suppose for distributions X_1, X_2, X_3 ... X_n with means \mu_1 , \mu_2 , \mu_3 ... \mu_n and standard deviation \sigma_1^2 , \sigma_2^2 , \sigma_3^2 ..... \sigma_n^2 respectively.
+\\
+Then X_1 + X_2 + X_3 will have mean *( \mu_1 + \mu_2 + \mu_3 + ... + \mu_n )* and standard deviation *(\sigma_1^2 + \sigma_2^2 + \sigma_3^2 + ..... + \sigma_n^2 )*
+
++ Additive Case
+Given,
+\[ X_1 \sim N(\mu_1, \sigma_1) \]
+\[ X_2 \sim N(\mu_2, \sigma_2) \]
+Then,
+\[ a X_1 + b X_2 \sim N \left(  a \mu_1 + b \mu_2, \sqrt{ a^2 \sigma_1^2 + b^2 \sigma_2^2} \right) \]

main.tsk

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+// *Remove Old
+// #do
+sh ./remove.sh
+
+*Emacs
+#do
+emacs --script export.el
+
+*LuaLatex
+#do
+lualatex main.tex