exp 1-4: init

Signed-off-by: Amneesh Singh <natto@weirdnatto.in>

file.org

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+#+LATEX_CLASS_OPTIONS: [a4paper,12pt]
+#+LATEX_HEADER: \usepackage[margin=0.5in]{geometry}
+#+LATEX_HEADER: \usepackage{fontspec}
+#+LATEX_HAEDER: \usepackage{graphicx}
+#+LATEX_HAEDER: \usepackage{}
+#+LATEX_HEADER: \setmainfont{LiberationSerif}
+#+LATEX_HEADER: \date{}
+#+OPTIONS: toc:nil
+#+OPTIONS: num:nil
+
+#+INCLUDE: front.tex
+#+LATEX: \clearpage
+
+#+INCLUDE: toc.tex
+#+LATEX: \clearpage
+
+* Experiment 1
+** Objective
+Installation of Scilab and demonstration of simple programming concepts like matrix multiplication (scalar and vector), loop, conditional statements and plotting.
+
+** Method
+1. Installed Scilab binary on desktop via nix package manager.
+2. Launched the binary and opened console.
+3. Declared matrix and integer, evaluated the product of matrix with a scalar
+
+  *Code*
+  #+begin_src
+  A = [4 8 12; 16 32 48; 64 72 80];
+  x = 5;
+  B = x * A;
+  disp(B);
+  #+end_src
+
+  *Output*
+  #+ATTR_LATEX: :width 3cm
+  [[./1a.png]]
+
+4. Declared another matrix and evaluated the vector product
+
+  *Code*
+  #+begin_src
+  A = [12 32 54; 9 4 2; 1 2 3];
+  B = [1 2 3; 7 2 4; 11 2 13];
+  x = 5;
+  C= x * A * B;
+  disp(C);
+  #+end_src
+
+  *Output*
+  #+ATTR_LATEX: :width 3cm
+  [[./1b.png]]
+
+5. Declared another matrix and evaluated the dot product
+
+  *Code*
+  #+begin_src
+  A = [12 32 54];
+  B = [1 2 3];
+  dot = A * B';
+  disp(dot);
+  #+end_src
+
+  *Output*
+  #+ATTR_LATEX: :width 3cm
+  [[./1c.png]]
+
+5. Declared another matrix and evaluated the dot product
+
+  *Code*
+  #+begin_src
+  x = 1:10; 
+  y = x .^ 2; 
+  plot(x,y); 
+  title(‘Square Function’); 
+  #+end_src
+
+  *Output*
+  #+ATTR_LATEX: :width 6cm
+  [[./1d.png]]
+
+** Result
+Installed Scilab and demonstrated simple programming concepts like matrix multiplication (scalar and vector), loop, conditional statements and plotting.
+
+#+LATEX: \clearpage
+* Experiment-2
+
+** Objective
+Program for demonstration of theoretical probability limits.
+
+** Method
+1. Opened Scilab console and evaluated the following commands.
+2. Declared integer n and set it to 10000, similarly declared and set another integer head_count to 0.
+3. Set up a loop from 1 to 10000 and generated a random number between 0 and 1 using rand() command
+4. if the value of number is less then 0.5 then incremented head_count by 1
+5. Set up a function P(i) for probability of heads in trial.
+6. Plotted the graph of P(i) using plot() command.
+
+** Code
+  #+begin_src
+n = 10000;
+head_count = 0;
+for i = 1:n
+  x = rand(1)
+
+  if x<0.5 then
+    head_count = head_count + 1;
+  end
+
+  p(i) = head_count / i;
+end
+
+disp(p(10000))
+
+plot(1:n,p)
+xlabel("No of trials");
+ylabel("Probability");
+title("Probability of getting Heads");
+#+end_src
+
+** Output
+  #+ATTR_LATEX: :width 6cm
+  [[./2.png]]
+
+** Result 
+
+#+LATEX: \clearpage
+* Experiment 3
+** Objective
+Program to plot normal distributions and exponential distributions for various parametric values.
+
+** a) Method: Normal Distribution
+1. Opened Scilab console and evaluated the following commands.
+2. Declared 2 arrays, m_values and s_values with various parametric values of mean and standard deviation.
+3. Set up a loop from 1 to length of the array ‘means’ and got a pair of values of mean and standard deviation.
+4. Using those values, generated the probability density function
+
+$f(\boldsymbol{x}) = \frac{1}{s}\cdot\sqrt{2\pi}\cdot e^{\frac{-(\boldsymbol{x} - m) ^ 2}{(2 * s ^ 2)}}$
+
+5. Created a range of x values to plot the normal distribution using linspace() command.
+5. Correctly titled and labelled the graph.
+6. Plotted various normal distribution curves using plot() command.
+
+** a) Code
+  #+begin_src
+m_values = [0, 1, -1];
+s_values =[0.5, 1, 1.5];
+
+for m = m_values
+  for s =s_values
+    t =grand(1, 1000, "nor", m, s);
+
+    x= linspace(m-4*s, m+4*s, 1000)
+    y = (1/s*sqrt(2*%pi))*exp(-(x - m).^2/(2*s^2));
+
+    plot(x, y);
+    hold on;
+    xgrid();
+  end
+
+end
+legend("m=0, s=0.5", "m=1, s=0.5", "m=-1, s=0.5",...
+       "m=0, s=1"  , "m=1, s=1"  , "m=-1, s=1", ...
+       "m=0, s=1.5", "m=1, s=1.5", "m=-1, s=1.5");
+
+xlabel("x")
+ylabel("Probability density function");
+title("Normal distributions for various parametric values");
+#+end_src
+
+** a) Output
+  #+ATTR_LATEX: :width 6cm
+  [[./3a.png]]
+
+** b) Method: Exponential Distribution
+1. Opened Scilab console and evaluated the following commands.
+2. Declared an array, Lambda_values with various values of lambda
+3. Set up a loop from 1 to length of the array Lambda_values and the function grand() is used to generate 1000 random numbers from the exponential distribution.
+4. Using those values, generated the probability density function
+
+$f(x) = Y = lambda \cdot e^{-lambda \cdot x}$
+
+5. Created a range of x values to plot the exponential distribution using linspace() command.
+5. Correctly titled and labelled the graph.
+6. Plotted various exponential distribution curves using plot() command.
+
+** b) Code
+  #+begin_src
+lambda_values = [0.5, 1, 2];
+for lambda = lambda_values
+  t = grand(1, 1000, "exp", lambda);
+  x = linspace(0, 8/lambda, 1000);
+  y = lambda * exp(-lambda * x);
+
+  plot(x, y);
+
+  xgrid();
+
+  hold on;
+end
+
+xlabel("x");
+ylabel("Probability density function");
+title("Exponential distributions for various values of lambda");
+
+legend(["lambda=0.5", "lambda=1", "lambda=2"]);
+#+end_src
+
+** b) Output
+  #+ATTR_LATEX: :width 6cm
+  [[./3b.png]]
+
+#+LATEX: \clearpage
+* Experiment 4
+** Objective
+Program to plot normal distributions and exponential distributions for various parametric values.
+
+** Theory 
+Binomial Distribution: A probability distribution that summarizes the likelihood that a variable will take one of two independent values under a given set of parameters. The distribution is obtained by performing a number of Bernoulli trials. A Bernoulli trial is assumed to meet each of these criteria: 
+1. There must be only 2 possible outcomes. 
+2. Each outcome has a fixed probability of occurring. A success has the probability of p, and a failure has the probability of 1 – p. 
+3. Each trial is completely independent of all others. 
+4. To calculate the binomial distribution values, we can use the binomial distribution formula:
+
+$P(X = x) = {}^{n}C_{x} \cdot p^x \cdot (1 - p)^{n - x}$
+
+where `n` is the total number of trials, `p` is the probability of success, and `x` is the number of successes. We can calculate the binomial distribution values for each possible value of `x` using this formula and the values of `n` and `p` given above. 
+ 
+** Problem statement
+6 fair dice are tossed 1458 times. Getting a 2 or a 3 is counted as success. Fit a binomial distribution and calculate expected frequencies. 
+ 
+** Method
+1. Find the number of cases, times the experiment is repeated, and the probability of success. 
+2. Here, we have to find the Binomial Probability Distribution, which is defined as: 
+
+$P(X = x) = \frac{n!}{x! \cdot (n-x)!} \cdot s^x \cdot (1 - s)^{n - x}$
+
+Calculate it. 
+
+3. Calculate the frequency, which is given by E= P*N. 
+4. Put the calculated values in the table below.
+
+|  x  | Expected Frequency | Binomial Distribution P (X = x) |
+|-----+--------------------+---------------------------------|
+| <c> |        <c>         |               <c>               |
+|  0  |       28.43        |            0.004831             |
+|  1  |       181.83       |            0.003107             |
+|  2  |       547.50       |            0.009387             |
+|  3  |      1009.53       |            0.017307             |
+|  4  |      1213.50       |            0.020803             |
+|  5  |       947.25       |            0.016255             |
+|  6  |       312.50       |            0.002132             |
+
+** Steps
+1. Find the number of cases, here, we take it as n. 
+2. Find the probability of success. Here, we take it as s (=2/6). 
+3. Define the number of times the process is repeated, and mark it as N. Here, acc to question, it’s 1458. 
+4. Take a variable x that varies from 0 to number of cases. 
+5. Apply the Formula for Binomial Probability Distribution.
+6. Apply the formula for the Frequency. 
+7. Plot the Graph.
+
+** Code
+  #+begin_src
+  n=6;
+  s=1/3;
+  N=1458;
+  x=0:n;
+  P= (1-s).^(n-x).*s.^x.*factorial(n)./(factorial(x).*factorial(n-x));
+  E= P*N;
+  clf();
+  plot(x,E,"b.-"); 
+  #+end_src
+
+** Output
+  #+ATTR_LATEX: :width 6cm
+  [[./4.png]]
+
+#+LATEX: \clearpage

front.tex

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+\newgeometry{left=1.5in,right=1.5in}
+\begin{titlepage}
+  \vspace*{1.2in}
+  \begin{center}
+    {\fontsize{20}{24}\selectfont \textbf{\emph{Probability and Statistics Lab}}}\\
+    BS-252
+  \end{center}
+  \vspace{0.3in}
+  \hspace{0.3in}
+  \begin{minipage}{2in}
+    Faculty Name:\\
+    Dr. Soumi Ghosh\\
+    Assisstant Professor\\
+    I.T. Department
+  \end{minipage}
+  \hfill
+  \begin{minipage}{2in}
+    Student: AMNEESH SINGH\\
+    Enrollment: 14114803121\\
+    Semester: IV\\
+    Group: I7
+  \end{minipage}
+
+  \begin{center}
+    \includegraphics[width=2in]{mait.png}\\
+    \fontsize{18}{22}\selectfont Maharaja Agrasen Institute of Technology, PSP Area, Sector-22, Rohini, New Delhi 110086 
+  \end{center}
+\end{titlepage}
+\restoregeometry

toc.tex

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+\begin{center}
+  \fontsize{15}{18}\selectfont \textbf{
+    PROBABILITY AND STATISTICS LAB\\
+    PRACTICAL RECORD
+  }
+\end{center}
+
+\begin{table}[h]
+  \begin{tabular}{lcl}
+    Paper Code & : & BS-252\\
+    Name of the student & : & Amneesh Singh\\
+    University Enrollment number & : & 14114803121\\
+    Branch & : & Information Technology\\
+    Group & : & I7
+  \end{tabular}
+\end{table}
+
+\textbf{PRACTICAL DETAILS}
+
+Experiments according to the lab syllabus prescribed by GGSIPU
+
+\begin{table}[h]
+ \fontsize{11}{12}\selectfont{
+    \renewcommand{\arraystretch}{2.5}
+    \begin{tabular}{|p{0.6cm}|p{8cm}|p{2cm}|p{2cm}|p{1cm}|} \hline
+      \textbf{Exp. No.} & \textbf{Experiment Name} & \textbf{Performance Date} & \textbf{Date Checked}& \textbf{Marks} \\ \hline \hline
+      & & & & \\ \hline
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+    \end{tabular}
+  }
+\end{table}