From 8552f74f917912f9a5161145728b72fdfd9a7130 Mon Sep 17 00:00:00 2001 From: lomna-dev Date: Fri, 28 Apr 2023 15:11:16 +0530 Subject: [PATCH] First Commit Basic Stuff --- build.sh | 4 + export.el | 2 + kurtosis.PNG | Bin 0 -> 60647 bytes kurtosis2.PNG | Bin 0 -> 116026 bytes main.org | 309 ++++++++++++++++++++++++++++++++++++++++++++++++++ main.pdf | Bin 0 -> 523050 bytes remove.sh | 13 +++ skewness.PNG | Bin 0 -> 127875 bytes 8 files changed, 328 insertions(+) create mode 100644 build.sh create mode 100644 export.el create mode 100644 kurtosis.PNG create mode 100644 kurtosis2.PNG create mode 100644 main.org create mode 100644 main.pdf create mode 100644 remove.sh create mode 100644 skewness.PNG diff --git a/build.sh b/build.sh new file mode 100644 index 0000000..1f9b897 --- /dev/null +++ b/build.sh @@ -0,0 +1,4 @@ +set -xe + +emacs --script export.el +lualatex main.tex diff --git a/export.el b/export.el new file mode 100644 index 0000000..01f7318 --- /dev/null +++ b/export.el @@ -0,0 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zk0w`$ZD)5ySwi!_^U(}X*0N-MwL8+t+1$wa^ifo5di>{i2W|;pBY^!I-9=snf~;^ zwV+E$QfFw>X0@~iko6?3a)Fv}>gN}polAnFW1;$=%fG@o6oWK1(;N4CnY=ASOu^XQ zuhqfqK|M^gGlK0xC zF@)IUS%7bLc7A@|5^^sL<^TRBjolsC@4GHGNG0IY>5$IK`R}cp5#TJ_f0o&5{!BgG zykT>VIYTU}8K{&7^?&B9^qU@z37YKdee5ot Kkt!j5|NjqIwNp3% literal 0 HcmV?d00001 diff --git a/main.org b/main.org new file mode 100644 index 0000000..25975b7 --- /dev/null +++ b/main.org @@ -0,0 +1,309 @@ +#+TITLE: Probability and Statistics ( BTech CSE ) +#+AUTHOR: Anmol Nawani + +# *Statistics + +* Ungrouped Data + +Ungrouped data is data that has not been arranged in any way.So it is just a list of observations + +\[ x_1, x_2, x_3, ... x_n \] + +** Mean +\[ \bar{x} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n} \] + +\[ \bar{x} = \frac{ \sum_{i = 1}^{n} x_i }{n} \] + +** Mode +The observation which occurs the highest number of time. So the x_i which has the highest count in the observation list. + +** Median +The median is the middle most observations. +After ordering the n observations in observation list in either Ascending or Descending order (any order works). The median will be : + ++ n is even + +\[ Median = \frac{ x_\frac{n}{2} + x_{(\frac{n}{2}+1)} }{2} \] + ++ n is odd + +\[ Median = x_\frac{n+1}{2} \] + +** Variance and Standard Deviation + +\[ Variance = \sigma^2 \] +\[ Standard\ deviation = \sigma \] + +\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - Mean)^2 }{n} \] + +\[ \sigma^2 = \frac{\sum_{i=1}^n x_i^2}{n} - (Mean)^2 \] + +** Moments + +*** About some constant A + +\[ r^{th}\ moment = \frac{1}{n} \Sigma(x_i - A)^r \] + +*** About Mean (Central Moment) + +When A = Mean, then the moment is called central moment. + +\[ \mu_r = \frac{1}{n} \Sigma(x_i - Mean)^r \] + +*** About Zero (Raw Moment) + +When A = 0, then the moment is called raw moment. + +\[ \mu_r^{'} = \frac{1}{n} \Sigma x_i^r \] + +* Grouped Data + +Data which is grouped based on the frequency at which it occurs. So if 9 appears 5 times in our observations, we group as x(observation) = 9 and f (frequency) = 5. + +#+attr_latex: :align |c|c|c| +|------------------+---------------| +| x (observations) | f (frequency) | +|------------------+---------------| +| 2 | 5 | +| 1 | 3 | +| 4 | 5 | +| 8 | 9 | +|------------------+---------------| + +If we store it in data way, i.e. the observations are of form 10-20, 20-30, 30-40 ... then we will get $x_i$ by doing + +\[ x_i = \frac{lower\ limit + upper\ limit}{2} \] + +i.e, + +$x_i$ for 20-30 will be $\frac{20 + 30}{2}$ + +So for data + +#+attr_latex: :align |c|c|c| +|-------+---------------| +| | f (frequency) | +|-------+---------------| +| 0- 20 | 2 | +| 20-40 | 6 | +| 40-60 | 1 | +| 60-80 | 3 | +|-------+---------------| + +the $x_i$'s will become. + +#+attr_latex: :align |c|c|c| +|-------+-----+-----| +| | f_i | x_i | +|-------+-----+-----| +| 0- 20 | 2 | 10 | +| 20-40 | 6 | 30 | +| 40-60 | 1 | 50 | +| 60-80 | 3 | 70 | +|-------+-----+-----| + + +** Mean + +\[ \bar{x} = \frac{ \Sigma f_i x_i}{\Sigma f_i } \] + +** Mode + +The *modal class* is the record with the row with the highest f_i + +\[ Mode = l + (\frac{f_1 - f_0}{2f_1 - f_0 - f_2}) \times h \] + +In the formula : \\ +l \rightarrow lower limit of modal class \\ +f_1 \rightarrow frequency(f_i) of the modal class \\ +f_0 \rightarrow frequency of the row preceding modal class \\ +f_2 \rightarrow frequency of the row after the modal class \\ +h \rightarrow size of class interval (upper limit - lower limit) + +** Median +The median for grouped data is calculated with the help of *cumulative frequency*. The cumulative frequency (cf_i) is given by: + +\[ cf_i = f_1 + f_2 + f_3 + ... + f_i \] + +The *median class* is the class whose cf_i is just greater than or is equal to $\frac{\Sigma f}{2}$ + +\[ Median = l + (\frac{(n/2) - cf}{f}) \times h \] + +In the formula : \\ +l \rightarrow lower limit of the median class \\ +h \rightarrow size of class interval (upper limit - lower limit) \\ +n \rightarrow number of observations \\ +cf \rightarrow cumulative frequency of the median class \\ +f \rightarrow frequency of the median class + +** Variance and Standard Deviation + +\[ Variance = \sigma^2 \] +\[ Standard\ deviation = \sigma \] + +\[ \sigma^2 = \frac{\sum_{i=1}^{n} f_i(x_i - Mean)^2 }{\Sigma f_i} \] + +\[ \sigma^2 = \frac{\sum_{i=1}^n f_ix_i^2}{\Sigma f_i} - (Mean)^2 \] + +** Moments + +*** About some constant A + +\[ r^{th}\ moment = \frac{1}{\Sigma f_i} [\Sigma f_i (x_i - A)^r] \] + +*** About Mean (Central Moment) + +When A = Mean, then the moment is called central moment. +\[ \mu_r = \frac{1}{\Sigma f_i} [\Sigma f_i (x_i - Mean)^r] \] + +*** About Zero (Raw Moment) + +When A = 0, then the moment is called raw moment. +\[ \mu_r^{'} = \frac{1}{\Sigma f_i} [\Sigma f_i x_i^r] \] + +* Relation between Mean, Median and Mode + +\[ 3Median = 2Mean + Mode \] + +* Relation between raw and central moments + +\[ \mu_0 = \mu_0^{'} = 1 \] +\[ \mu_1 = 0 \] +\[ \mu_2 = \mu_2^{'} - \mu_1^{'2} \] +\[ \mu_3 = \mu_3^{'} - 3\mu_1^{'}\mu_2^{'} + 2\mu_1^{'3} \] +\[ \mu_4 = \mu_4^{'} - 4\mu_3^{'}\mu_1^{'} + 6\mu_2^{'}\mu_1^{'2} - 3\mu_1^{'4} \] + +* Skewness and Kurtosis + +** Skewness + ++ If Mean > Mode, then skewness is positive ++ If Mean = Mode, then skewness is zero (graph is symmetric) ++ If Mean < Mode, then skewness is zero + +[[./skewness.PNG]] + +*** Pearson's coefficient of skewness + +The pearson's coefficient of skewness is denoted by S_{KP} + +\[ S_{KP} = \frac{Mean - Mode}{Standard\ Deviation} \] + ++ If S_{KP} is zero then distribution is symmetrical ++ If S_{KP} is positive then distribution is positively skewed ++ If S_{KP} is negative then distribution is negatively skewed + +*** Moment based coefficient of skewness + +The moment based coefficient of skewness is denoted by \beta_1. The \mu here is central moment. + +\[ \beta_1 = \frac{\mu_3^2}{\mu_2^3} \] + +The drawback of using \beta_1 as a coefficient of skewness is that it *can only tell if distribution is symmetrical or not* ,when $\beta_1 = 0$. +It can't tell us the direction of skewness, i.e positive or negative. + ++ If \beta_1 is zero, then distribution is symmetrical + +*** Karl Pearson's \gamma_1 + +To remove the drawback of the \beta_1 , we can derive Karl Pearson's \gamma_1 + +\[ \gamma_1 = \sqrt{\beta_1} \] +\[ \gamma_1 = \frac{\mu_3}{\mu_2^{3/2}} \] + ++ If \mu_3 is positive, the distribution has positive skewness ++ If \mu_3 is negative, the distribution has negative skewness ++ If \mu_3 is zero, the distribution is symmetrical + +** Kurtosis + +Kurtosis is the measure of the peak and the curve and the "fatness" of the curve. + +# https://www.analyticsvidhya.com/blog/2021/05/shape-of-data-skewness-and-kurtosis/ +[[./kurtosis.PNG]] + +# https://www.bogleheads.org/wiki/Excess_kurtosis +[[./kurtosis2.PNG]] + +The kurtosis is calculated using \beta_2 + +\[ \beta_2 = \frac{\mu_4}{\mu_2^2} \] + +The value of \beta_2 tell's us about the type of curve + ++ Leptokurtic (High Peak) when \beta_2 > 3 ++ Mesokurtic (Normal Peak) when \beta_2 = 3 ++ Platykurtic (Low Peak) when \beta_2 < 3 + +*** Karl Pearson's \gamma_2 + +\gamma_2 is defined as: + +\[ \gamma_2 = \beta_2 - 3 \] + ++ Leptokurtic when \gamma_2 > 0 ++ Mesokurtic when \gamma_2 = 0 ++ Platykurtic when \gamma_2 < 0 + +# *Probability + +* Basic Probability + +** Conditional Probability + +If some event B has already occured, then the probability of the event A is: + +\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} \] + +$P(A \mid B)$ is read as A given B. So we are given that B has occured and this is probability of now A occuring. + +** Law of Total Probability + +The law of total probability is used to find probability of some event A that has been partitioned into several different places/parts. + +\[ P(A) = P(A|B_1)P(B_1) + P(A|B_2)P(B_2) + P(A|B_3)P(B_3) + ... + P(A|B_i)P(B_i) \] +\[ P(A) = \Sigma P(A|B_i)P(B_i) \] + +*Example*, Suppose we have 2 bags with marbles + ++ Bag 1 : 7 red marbles and 3 green marbles ++ Bag 2 : 2 red marbles and 8 green marbles + +Now we select one bag at random (i.e, the probability of choosing any of the two bags is equal so 0.5). If we draw a marble, what is the probability that it is a green marble? + +*Sol.* The green marbles are in parts in bag 1 and bag 2. \\ +Let G be the event of green marble. \\ +Let B_1 be the event of choosing the bag 1 \\ +Let B-2 be the event of choosing the bag 2 \\ + +Then, $P(G|B_1) = \frac{3}{7 + 3}$ and $P(G|B_2) = \frac{8}{2 + 8}$ +\\ +Now, we can use the law of total probability to get + +\[ P(G) = P(G|B_1)P(B_1) + P(G|B_2)P(B_2) \] + +*Example* 2, Suppose a there are 3 forests in a park. ++ Forest A occupies 50% of land and 20% plants in it are poisonous ++ Forest B occupies 30% of land and 40% plants in it are poisonous ++ Forest C occupies 20% of land and 70% plants in it are poisonous +What is the probability of a random plant from the park being poisonous. + +*Sol.* Since probability is equal across whole area of the park. Event A is plant being from Forest A, Event B is plant being from Forest B and Event C is plant being from Forest C. If event P is plant being poisonous, then using law of total probability, + +\[ P(P) = P(P|A)P(A) + P(P|B)P(B) + P(P|C)P(C) \] + +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) = 0.40 and P(P|C) = 0.70 + + +** Some basic identities + ++ Probabilities follow law of inclusion and exclusion +\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] + ++ DeMorgan's Theorem +\[ P(\overline{A \cap B }) = P(\overline{A} \cup \overline{B}) \] +\[ P(\overline{A \cup B }) = P(\overline{A} \cap \overline{B}) \] + ++ Some other Identity +\[ P(\overline{A} \cap B) + P(A \cap B) = P(B) \] +\[ P(A \cap \overline{B}) + P(A \cap B) = P(A) \] diff --git a/main.pdf b/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0aa9fec07d875d50aae44b2de6b26c7c25d692dd GIT binary patch literal 523050 zcma&N18^=uw=Nv#jcxDP$&PJXJGO1xJGO1xwrx8*ww*uUy?@ocRp->LGc{G+H9g&H zYE94cEDfo=uqZ7f9Sbz+shKYjvK6qUFWJd;xd+~1Q7`B z34vLKp_7dX(Mw91prmu?nqktERFYrb1OP==ER9A^>^raHikCa*)`oOWNJM;pTC$TC;8}k-AGuG14uPs-f_d2lkH20t-%x8gB_GE z#%Z~G4+zShJu{^sY1|La#}*k)jSI&ZGUP4aipDjSPJL`}w93)Jfx#I?M4#q~Sh!oo z9j8ZEcTcU=CzJl3({Mj^t!9B^4Y=h82B+#mdHwyS!mqfy0q+yZeEm?z>ybKdoB$IE 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