Question

Suppose that y = x², where x is a normally distributed random variable with a mean and variance of µ_x = 0 and σ²_x = 4. Find the mean and variance of y by simulation. Does µ_y = µ²_x? Does σ_y = σ²_x? Do this for 100, 1000, and 5000 trials.

Answer 1

Program plan:

• To find the mean and variance of the given equation where the variable is normally distributed.

 

Program:

%**********************************************************

%A matlab code is written to find the mean and variance of

%the given equation where the variable is the normally

%distributed number with mean 0 and variance 4.

%**********************************************************

%Declaration of constant

m=100;

%Declaration of 'm' random numbers

x=2.*randn(1,m);

%Computes the value

y=x.^2;

%Computes the mean of array

mu_y=mean(y);

%Displays text

disp('The mean of y is ');

%Displays value

disp(mu_y);

%Computes the standard deviation of the array

variance_y=std(y)^2;

%Displays text

disp('The variance of y is');

%Displays value

disp(variance_y);

 

Output:

The output is found for three values of ‘m’.

m=100;

 

>> Untitled

The mean of y is

3.9944

The variance of y is

37.7325

 

m=1000;

>> Untitled

The mean of y is

3.9331

The variance of y is

29.0860

 

m=5000;

>> Untitled

The mean of y is

4.0030

The variance of y is

31.4172

The output is found for three values of ‘m’.

m=100;

 

>> Untitled

The mean of y is

3.9944

The variance of y is

37.7325