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z?b?0&_D!5PF-C9t(Iin35m1NO$EW7c$K!Hxaye$R8&pGLV(whJWMpG==f;hQ*jQT= zlPx7LgVZKR1_T5I2QR*wWo&GmoRoCu?%mzHcD=i^^YW!jhaP`4%em1|{eRWEb?a8G zy7c<%jphGl&6;&>eZ0JgNJwO4l~ATV5WcH~nmyxR_X2SlBF|WwE=)pZ;i^$%F2p~xickOaKGQ$wQG|$T3A_CRaE@=e!o6HD@$pj zhkVV4#_ik9yN@1PSN7%GH^2Rd4@hx=`F=3`)J zc-)-HPN@HZOWQ*qKR>D7V<%6Zd_K=`_F0QG)&pzUF7q)QTDgAx{oUp7@9r+2Uz_&* z-QA^2mm(xNxw*al{q0L%UAa7Y=bbxuwrtt*;zdS{FmuAUrc8E*hZkOcdGH`%rcc{i z+se<+e8sv&B_vKfEcoEZ+i^O zeED)`>FcnptSlcNp84nPi=X)?wW%1Q4c6Rp7n>KkYseivPGBVO?uHWK|4`xd+Y!;6> z$ | | | | | +| No. of Heads ($x$) | 0 | 1 | 2 | 3 | +| Frequency ($f$) | 36 | 40 | 22 | 2 | + +Fit a binomial distribution and calculate expected frequencies. + +** Method +1. For given data, total number of coins($n$) = 3 and total number of trials($N$) would be = 100 +2. Find $X \cdot F$ to calculate the mean, for all corresponding values of $X$ and $F$, and calculate $\Sigma F \cdot x$ +Mean would then be = $\frac{\Sigma F \cdot x}{n}$ i.e., $\frac{90}{100} = 0.9$. + +3. Since we know for binomial distribution, $mean = n \cdot p$. So, $0.9 = 3 \cdot p$ => $p = 0.3$. +4. Calculate variance. We already know that $var = n \cdot p \cdot (1-p)$. +5. Now, calculate Binomial distribution, $P(X= x)$ and calculate expectes valued using $P \cdot N$. + +| | $X$ | $F$ | $X \cdot F$ | +|---+------------+-----+-------------| +| | | | | +| / | < | < | <> | +| | 0 | 36 | 0 | +| | 1 | 40 | 40 | +| | 2 | 22 | 44 | +| | 3 | 2 | 6 | +| | TOTAL($N$) | 100 | 90 | + +| $X$ | 0 | 1 | 2 | 3 | TOTAL | +|--------------------+-----+------+------+-----+-------| +| | | | | | | +| Observed Frequency | 26 | 40 | 2 2 | 2 | 100 | +| Expected Frequency | 343 | 44.1 | 18.9 | 2.7 | 100 | + +** Code +#+begin_src +n = 3; +N = 100; +F = [36, 40, 22, 2]; +X = [0,1,2,3]; + +FX = (36*0+40*1+22*2+2*3); +Mean = FX/N; +disp("Mean = ", Mean); + +p = Mean/n; +disp("Probability of success = ",p); + +var = n*p*(1-p); +disp("Variance = ",var); + +x = 0:3; +P= (1-p).^(n-x).*p.^x.*((factorial(n))./((factorial(x).*factorial(n-x)))); +disp("Binomial distribution = ",P); + +E = N*P; +disp("Expected Frequency = ", E); + +clf(); +plot(X,E,"b.-"); + +xlabel('X '); +ylabel('Expected Frequency'); +title('Binomial Distribution '); + +Y = grand(1, 80, "bin", 3,p); +disp(Y); + +histplot(30, Y, normalization=%f, style=5); +#+end_src + + +** Output +#+ATTR_LATEX: :width 10cm +[[./5.png]] + +#+LATEX:\clearpage +* Experiment 6 +** Objective +Program for fitting of Poisson distributions for given value of lambda.. + +** Theory +Poisson Distribution: The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a known constant rate and independently of the time since the last event. It is often used to model rare events. + +The Poisson distribution has only one parameter, often denoted as $\lambda$, which represents the average rate of occurrence of the events over the given interval. The probability of observing exactly k events in the interval is given by the Poisson probability mass function: + +$P(X=k) = (e^{-\lambda} * \lambda^k) / k!$ + +where $X$ is the random variable representing the number of events, $e$ is the base of the natural logarithm, and $k!$ denotes the factorial of $k$. + + +The mean and variance of the Poisson distribution are both equal to $\lambda$, which means that the distribution is unimodal and symmetric around $\lambda$. + +The Poisson distribution is also a limiting case of the binomial distribution, when the number of trials $n$ goes to infinity and the probability of success $p$ goes to zero, but the product $n \cdot p$ remains constant and equal to $\lambda$. + +** Problem statement +A set of three similar coins are tossed 100 ($N$) times with the following results in Table 1. Note the difference with experiment number 4 where “observed frequencies” were not given. + +#+CAPTION: Table 1 +| | | | | | +| / | < | | | | +| No. of Heads ($x$) | 0 | 1 | 2 | 3 | +| Frequency ($f$) | 36 | 40 | 22 | 2 | + +Fit a binomial distribution and calculate expected frequencies. + +** Method +1. For given data, create a table (Table 2) to record the values of $x$, $f$, and $xf$. Here, $x$ denotes the number of heads obtained in a single toss, $f$ is the frequency of obtaining $x$ heads, and $xf$ is the product of $x$ and $f$. + +#+CAPTION: Table 2 +| | | | | | +| / | < | | | | +| $x$ | 0 | 1 | 2 | 3 | +| $f$ | 36 | 40 | 22 | 2 | +| $xf$ | 0 | 40 | 44 | 6 | + +2. [@2] The mean of the binomial distribution can be calculated as $N \cdot p$, where $N$ is the total number of trials and $p$ is the probability of getting a head. Here, $N=100$. +3. Calculate the value of $p$, using the formula $p = \frac{x}{N}$, where $x$ is the number of heads obtained in the 100 tosses. So, $p = (036 + 140 + 222 + 32)/100 = 0.64$. +4. Using the formula for the binomial distribution, calculate the expected frequency of obtaining x heads in a single toss as: +$P(X = x) = {}^{n}C_{x} \cdot p^x \cdot (1 - p)^{n - x}$ +#+OPTIONS: \n:t + +Where $n$ is the total number of trials, $x$ is the number of successful trials, and ${}^{n}C_{x}$ is the binomial coefficient. We can then multiply this value by 100 to get the expected frequency for each value of $x$. + +Tabulate this information in Table 3. + +#+CAPTION: Table 3 +| | | | | | | +| / | < | | | | | +| $X$ | 0 | 1 | 2 | 3 | Total | +| Observed Frequency | 36 | 40 | 22 | 2 | 100 | +| Expected Frequency $100 \cdot P(X=x)$ | 23.65 | 37.84 | 27.51 | 10.00 | 100.00 | + +** Code +#+begin_src +observed_freq = [36, 40, 22, 2]; +N = sum(observed_freq); +x = 0:3; + +mean = sum(x .* observed_freq) / N; + +// Poisson distribution with lambda = mean +expected_freq = N * exp(-mean) * (mean .^ x) ./ factorial(x); + +// Output tables +disp(["x", "Observed Freq", "Expected Freq"]); +disp([x', observed_freq', expected_freq']); + +// Plot of expected frequencies +clf(); +plot(x, expected_freq, '.-', 'LineWidth', 2); +xlabel('Number of Heads'); +ylabel('Expected Frequency'); +title('Fitting of Poisson distribution'); +legend('Expected Frequencies'); +#+end_src + +** Output +#+ATTR_LATEX: :width 10cm +[[./6.png]] + +#+LATEX: \clearpage +* Experiment 7 + +** Objective: +Plotting Regression line for the given data points. + +** Formulation and Method +*** Regression line +\begin{equation} + y = m \cdot x + b +\end{equation} + +where, $m$ = slope, $b$ = y-intercept. + +*** Method of least squares: +\begin{equation} + E = \Sigma_i (Y_i – (m \cdot x +b))^2 +\end{equation} +Find $\frac{\partial e}{\partial m}, \frac{\partial e}{\partial b} = 0$ + +*** Problem statement +Find the regression line for the given data points (x,y) + +| X | Y | +|-----+------| +| | | +| 20 | 0.18 | +| 60 | 0.37 | +| 100 | 0.35 | +| 140 | 0.70 | +| 160 | 0.56 | +| 220 | 0.75 | +| 260 | 0.18 | +| 300 | 0.36 | +| 340 | 1.17 | +| 380 | 1.65 | + +** Method +1. Make $x$ data and $y$ data lists. +2. Use regline() to find $m$ and $b$ +3. Use scatter() to plot $x$ data and $y$ data +4. Use plot() to draw regression line. + + +** Results +The following are required as output: +1. The full code. +2. The plot for regression line and data points. + +** Code +#+begin_src +x_data = [20,60,100,140,160,220,260,300,340,380] +y_data = [0.18,0.37,0.35,0.70,0.56,0.75,0.18,0.36,1.17,1.65] + +[a, b] = reglin(x_data, y_data); +scatter(x_data,y_data,30,"x") +plot(x_data, a*x_data+b,"red") + +xlabel("X") +ylabel("Y") +title("Simple Linear Regression") +#+end_src + +** Output +#+ATTR_LATEX: :width 10cm +[[./7.png]] diff --git a/file.pdf b/file.pdf index facb7a29912bd17f7b472bfca44a5c5312331987..1b591fff4147fffda3f09c6e62abfacb64d043ce 100644 GIT binary patch delta 156445 zcmdmWL~{B%*$r=c8O^6FzGD`tUpw=3-)%Pm*YCf${vFuzJM=FD3zO7Qt_QIKZ+X@y zsmwm3wE9oos(9C+B{CK7bU6A>th*ihy+qt{(!77OzI=J@X>Ymzp8xzjm&qTss;3_= z`TsfPN8h|NDNZ4t%Qe%MZ(98FpINPK_FEbGBkfmjeY*LpZr`WE>~!&;JI?91J=)n@ zU(j~Gu&#U)?g$-KOil$8_lZ`;%N2LkQp@2;Ag`>*W3Qf1WSsor^Ah8p{R z&PeaAxUlAW-kkVv&+W9n25zhGEA4xDdHL+KlMS_mXYMdN`FZOJxt&WU-LH63$)}Yb zrY1FYx@E`n4;$~yj1MX}s`Qy#eQ~L*x*#dfkN20w%#L62GkDVJc~&Qt;@PG&mKJDC%HH(vmV3;rxr@(~ 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