Question

Please solve using MATLAB. Will rate

Problem 4; Applications of functions in numerical computations The probability of occurrence based on a normalized Gaussian distribution requires the evaluation of the following integral with the aid of the built-in function integral, plot the above function o the domain 0szs5. Plot on the same graph the above function by evaluating the integral using the RightRiemannSum function. Critical thinking: For this problem, it might be easier to use the plot function than the fplot function. But either approach will work. To use plot, one needs to generate a set of data points of the function versus the discretized z values. To use fplot, one needs to properly define the probability function

Answer 1

The MATLAB code for plotting both curves is given below. It contains the usage of another functioned that is used to calcuate the Right Riemann Sum which is to be defined in another file named rsum_1.m . The corresponding code is given further below. The resulting plot is attached.

fun=@(x)exp(-x.^2/2); %function defined in terms of variable x
k=1;
f=zeros(1,21);
g=zeros(1,21);
for z=0:0.25:5 %step-size taken as 0.25 to plot enough number of points
f(k)=integral(fun,0,z)/sqrt(2*pi); %gaussian distribution evaluated using 'integral'
g(k)=rsum1(fun,0,z,20)/sqrt(2*pi); %gaussian distribution evaluated using Right Riemann Sum
k=k+1;
end
xaxis=linspace(0,5,21); %generate 21 equally spaced points between 0 and 5
plot(xaxis,f) % plot the gaussian curve(integral) against this vector
hold on; % hold on to plot another curve
plot(xaxis,g) % plot gaussian curve (riemann sum) against this vector
xlabel('z'); %define labels of the axes
ylabel('f(z)');

rsum_1.m

function value=rsum_1(Fun,a,b,number)
%a and b are the lower and upper limits of function to be approximated reepectively
%number is step-size of approxiamting the integral(number of rectangles)
delx=(b-a)/number; %width of the rectangle
Rsum=0;
for j=1:number
Rsum=Rsum+vpa(feval(Fun,a+(j)*delx))*delx; %evaluates the function at right-end point (a+dx,a+2dx,a+3dx...) %and adds it to resuls
end
value=Rsum;
end

The blue curve corresponds to the function evaluated using integral and the red curve belongs to the function evaluated using Right Riemann Sum.

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.5 1.5 2.5 3 3.5 4.5