Question

Question:

Determine The Probability Density Function For The Following Cumulative Distribution Function. 

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Answer 1

$f_x(x; \sigma^2)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{x^2}{2 \sigma^2}}$

By factorization theorem , we can write it as

$=g(\sum_i^n x_i^2 \sigma^2)*h(x)$

Therefore

$T=\sum_1^n x_i^2$ is sufficient statistic for $\sigma^2$ where $h(x)=1$

% Clearing Screen
clc

% Setting x as symbolic variable
syms x;

% Input Section
y = input('Enter non-linear equations: ');
a = input('Enter initial guess: ');
e = input('Tolerable error: ');
N = input('Enter maximum number of steps: ');
% Initializing step counter
step = 1;

% Finding derivate of given function
g = diff(y,x);

% Finding Functional Value
fa = eval(subs(y,x,a));

while abs(fa)> e
    fa = eval(subs(y,x,a));
    ga = eval(subs(g,x,a));
    if ga == 0
        disp('Division by zero.');
        break;
    end
    
    b = a - fa/ga;
    fprintf('step=%d\ta=%f\tf(a)=%f\n',step,a,fa);
    a = b;
    
    if step>N
       disp('Not convergent'); 
       break;
    end
    step = step + 1;
end

fprintf('Root is %f\n', a);