From 9fe80885bfe3d4af04085f0f7a4a92095456d78b Mon Sep 17 00:00:00 2001 From: lomna-dev Date: Sun, 14 May 2023 00:44:01 +0530 Subject: [PATCH] Till Normal Distribution TODO : Standard normal distribution funtion Z --- main.org | 269 +++++++++++++++++++++++++++++++++++++++++++++++++++++++ main.pdf | Bin 523050 -> 581836 bytes main.tsk | 11 +++ 3 files changed, 280 insertions(+) create mode 100644 main.tsk diff --git a/main.org b/main.org index 25975b7..0d4178f 100644 --- a/main.org +++ b/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, +\[ a X_1 + b X_2 \sim N \left( a \mu_1 + b \mu_2, \sqrt{ a^2 \sigma_1^2 + b^2 \sigma_2^2} \right) \] diff --git a/main.pdf b/main.pdf index 0aa9fec07d875d50aae44b2de6b26c7c25d692dd..83d1d1c14dfa765c552386d156a7c659c3fafce7 100644 GIT binary patch delta 293726 zcmZsCb8sfk(r#>RY;Wwmv2EM7?YtY?+Ss;j+t}E)H@v@d?)h%rs_#_IOmz+ZnVRnB z>F()@XQHWovUv2g_XB7uFcyxqx*|wI2xe9ejwB&Sbl{fOhCLn!ir-G{u8v7wnO84} z&@VvEXM7PNYAY37uVxyAuw5<5xctKlUt@h+L8{OEusCYpv9iqdRF+Gqi1klwi%{wr zI<$ehzTiFC-B}q5^rQji{PZPkM_KO}(2}BI5EXTFWtKuF!{1Jcdp#Hd<9$`Irb}LH 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