Question

2. Suppose the probability density function for American male height is roughly (in inches x) h(x)e-a-69)2/5.6 2.8V2 a) Write a code to estimate the probability that an American male is between 65 and 70 inches tall using the following integration methods i. multiple application of Trapezoidal rule with 2 and 4 ii, multiple application of Simpson's 1 /3 rule with n = 2 and n = 4 iii. single application of Simpson's 3/8 rule b) Write a code to estimate the probability that an American male is over 65 inches tall using single application of Simpson's 1/3 rule.

Answer 1

Matlab code for finding integration using different method clear all close all %Probability distribution function h-e (x) (1./(2.8.*sqrt (2 pi)))*exp( (-(x-69).2)/5.6) %height between 65 to 70 a 65; b-70; n-2; %Function for which integration have to do fprintfnProbability distribution function h(x) \n') disp (h) fprint f ("Height between [%2.2f to %2.2f] is \n', a,b) %integration using Simpson 1/3 and Simpson 3/8 method val_simp 13-Simp13_int (h,a,b,n) val_simp38-Simp38 int(h,a,b,n) val-trap =trap-int ( h, a,b,n); fprint f("\tIntegration using Simpson 1/3 for 2 interval is %f. n',val_simp13) fprint f ( -\tIntegration using Simpson 3/8 for 2 interval IS . n,val_simp38) tprint f("\tIntegration using Trapi zoidal method for 2 interval is %f. n',val_trap) %height between 65 to 70 a-65; b-70; n-4; Function for which integration have to do fprintf (nProbability distribution function h(x)n) disp (h) fprintf("Height between [%2.2f to %2.2f] is \n", a,b) %integration using Simpson 1/3 and Simpson 3/8 method val simp 13-Simp13_int(h, a,b, n); val simp 38-Simp38_int(h, a,b,n); valtrap -trap int (h, a,b,n); fprint f ('\tIntegration using Simpson 1/3 for 4 interval is %f. n',val_simp13) fprint f("1tIntegration using Simpson 3/8 for 4 interval is %f. n,val_simp38) fprintf("AtIntegration using Trapizoidal method for 4 interval is %f. n,val_trap) %height between 65 to 70 a-65; b-1000; n-4000; %Function for which integration have to do fprintf (nProbability distribution function h(x)n) disp (h) fprintf("Height greater than %d is \n',a)

%integration using Simpson 1/3 and Simpson 3/8 method val simp 13-Simp13_int(h, a,b,n); val-simp 38-sinp38-int (h , a,b, n) ; valtrap -trap int (h, a,b,n); fprint f ("AtIntegration using Simpson 1/3 for 4 interval is %f. n,val simp13) fprint f("1tIntegration using Simpson 3/8 for 4 interval is %f. n,val_simp38) fprintf("AtIntegration using Trapizoidal method for 4 interval is %f. n,val_trap) 88응888888888888888888888888888888888888888888888888888888 % Matlab function for mid point integration function val-trap int(f,a,b,N) % func is the function for integration % a is the lower limit of integration % b is the upper limit of integration % N number of rectangles to be used val-0; splits interval a to b into N+1 subintervals xx-linspace (a, b, N+1) dx=xx ( 2 )-xx (1); %x interval %loop for Riemann integration for i-2: length (xx)-1 val-val+dx*double(fxx1)); end val-val+dx* (0.5*double(f (xx (1)))+0.5*double(f (xx(end))); end Matlab function for Simpson 113 Method function val-simp13_int(f,a,b,n) f-function for which integration have to do a-upper limit of integration %h-lower limit of integration %n-number of subintervals zs-f (a ) + f (b); %simpson integration a 11 x values for given subinterval xx-1 in space ( a, b,n+1 ) ; dx-( xx ( 2 )-xx ( 1 ) ) ; %x interval simpson Algorithm for n equally spaced interval for i=2:n if mod(i,2)-0 else zs-zs+2*f (xx(i)); end end result using Simpson rule val-double ((dx/3) *zs)

end %%Matlab function forsimpson 3/8 Method function val-Simp38_int(f,a,b,n) f is the function for integration % a is the lower limit of integration % b is the upper limit of integration % n is the number of trapi zoidal interval in [a,b] splits interval a to b into n+l subintervals xx-linspace (a,b, n+1) dx-xx ( 2 )-xx ( 1 ) ) ; %x interval val=f(a)+f (b); %loop for trapizoidai integration for i=2:n if mod ( i-1,3,-#0 val-val+2*double (f (xx() else val-val+3*double (f (xx(i)) end result using midpoint integration method val- (3/8) *dx*val; 응욤욤응응욤욤各各욤욤各各욤욤各各욤 End of Code 各各음음各各음음各各음음各各음음各 Probability distribution function h(x)- (x) (1./(2.8.*sqrt (2 pi) exp((-(x-69). 2)/5.6) Height between 65.00 to 70.00j is Integration using Simpson 1/3 for 2 interval is 0.423923. Integration using Simpson 3/8 for 2 interval is 0.387535 Integration using Trapizoidal method for 2 interval is 0.397543 Probability distribution function h(x)- (x) (1.1/(2.8.*sgrt (2*pi))) exp((-(x-69). 2)/5.6) Height between 65.00 to 70.00J is Integration using Simpson 1/3 for 4 interval is 0.428878. Integration using Simpson 3/8 for 4 interval is 0.377778 Integration using Trapizoidal method for 4 interval is 0. 421044. Probability distribution function h(x)= (x) (1./(2.8.*sqrt (2 *pi))) exp((-(x-69). *2)75.6) Height greater than 65 is Integration using Simpson 1/3 for 4 interval is 0.592586. Integration using Simpson 3/8 for 4 interval is 0.592586. Integration using Trapizoidal method for 4 interval is 0.592533.

%Matlab code for finding integration using different method
clear all
close all
%Probability distribution function
h=@(x) (1./(2.8.*sqrt(2*pi)))*exp((-(x-69).^2)/5.6);
%height between 65 to 70
a=65; b=70; n=2;
%Function for which integration have to do
fprintf('\nProbability distribution function h(x)=\n')
disp(h)
fprintf('Height between [%2.2f to %2.2f] is \n',a,b)

%Integration using Simpson 1/3 and Simpson 3/8 method
    val_simp13=Simp13_int(h,a,b,n);
    val_simp38=Simp38_int(h,a,b,n);
    val_trap =trap_int(h,a,b,n);
fprintf('\tIntegration using Simpson 1/3 for 2 interval is %f.\n',val_simp13)

fprintf('\tIntegration using Simpson 3/8 for 2 interval is %f.\n',val_simp38)

fprintf('\tIntegration using Trapizoidal method for 2 interval is %f.\n',val_trap)

%height between 65 to 70
a=65; b=70; n=4;
%Function for which integration have to do
fprintf('\nProbability distribution function h(x)=\n')
disp(h)
fprintf('Height between [%2.2f to %2.2f] is \n',a,b)

%Integration using Simpson 1/3 and Simpson 3/8 method
    val_simp13=Simp13_int(h,a,b,n);
    val_simp38=Simp38_int(h,a,b,n);
    val_trap =trap_int(h,a,b,n);
fprintf('\tIntegration using Simpson 1/3 for 4 interval is %f.\n',val_simp13)

fprintf('\tIntegration using Simpson 3/8 for 4 interval is %f.\n',val_simp38)

fprintf('\tIntegration using Trapizoidal method for 4 interval is %f.\n',val_trap)

%height between 65 to 70
a=65; b=1000; n=4000;
%Function for which integration have to do
fprintf('\nProbability distribution function h(x)=\n')
disp(h)
fprintf('Height greater than %d is \n',a)

%Integration using Simpson 1/3 and Simpson 3/8 method
    val_simp13=Simp13_int(h,a,b,n);
    val_simp38=Simp38_int(h,a,b,n);
    val_trap =trap_int(h,a,b,n);
fprintf('\tIntegration using Simpson 1/3 for 4 interval is %f.\n',val_simp13)

fprintf('\tIntegration using Simpson 3/8 for 4 interval is %f.\n',val_simp38)

fprintf('\tIntegration using Trapizoidal method for 4 interval is %f.\n',val_trap)

     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Matlab function for mid point integration
function val=trap_int(f,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    val=0;
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            val=val+dx*double(f(xx1));
        end
       val=val+dx*(0.5*double(f(xx(1)))+0.5*double(f(xx(end))));
end

%%Matlab function for Simpson 1/3 Method
function val=Simp13_int(f,a,b,n)
%f=function for which integration have to do
%a=upper limit of integration
%b=lower limit of integration
%n=number of subintervals

    zs=f(a)+f(b);   %simpson integration
    %all x values for given subinterval
    xx=linspace(a,b,n+1);
    dx=(xx(2)-xx(1)); %x interval
    %Simpson Algorithm for n equally spaced interval
    for i=2:n
        if mod(i,2)==0
            zs=zs+4*f(xx(i));
        else
            zs=zs+2*f(xx(i));
        end
    end
    %result using Simpson rule
    val=double((dx/3)*zs);
end

%%Matlab function forSimpson 3/8 Method
function val=Simp38_int(f,a,b,n)
    % f is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % n is the number of trapizoidal interval in [a,b]
  
    %splits interval a to b into n+1 subintervals
    xx=linspace(a,b,n+1);
    dx=(xx(2)-xx(1)); %x interval
    val=f(a)+f(b);
    %loop for trapizoidal integration
        for i=2:n
            if mod(i-1,3)==0
                val=val+2*double(f(xx(i)));
            else
                val=val+3*double(f(xx(i)));
            end
        end
    %result using midpoint integration method
    val=(3/8)*dx*val;
end
  
%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%