Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 4.2 Continuous Density Functions
In this example we will compute the cumulative distribution function. Let us begin with the
density function over a range 0 <x < b and 0 < y <

syms u v b C
pretty (C*exp(-u-v))
C exp(-u - v)
The normalization constant is obtained by integrating over the range and setting it equal to 1 or
FI=int(int(C*exp(-u-v),u,0,b),v,0,inf);

Performing the integrations using Matlab and solving for C we have
C=solve(FI-1,C)
C =
-1/(exp(-b)-1)
The cumulative distribution function can be found by evaluating the expression
syms x y CT
F=int(int(CT*exp(-u-v),u,0,x),v,0,y);
F=simplify(F);
F=expand(F);
prettY(F)
CT
CT
CT
------------- - ------ - ------ + CT
exp(y) exp(x)
exp(x)
exp(y)
We may indeed verify that the density function results by differentiating F(x,y) with respect to x and
y as
f=diff(diff(F,x),y)
f =
CT/exp(y)/exp(x)
The marginal distributions can be found by integrating over the x range for f(y) and the y range for
f(x)
f=subs(f,x,u);
fy=int(f,u,0,b)
fy =
-CT*exp(-y-b)+CT*exp(-y)
It is possible to reduce the expression by substitution for C and simplifying. Some rearranging and
factoring is needed to help Matlab recognize the simplifications
fy=subs(fy,CT,C);
pretty(fy)
exp(-y - b)
exp(-y)
----------- - ----------exp(-b) - 1
exp(-b) - 1
fy=expand(fy);
fy=factor(fy);
pretty(fy)
1
-----exp(y)
The range of y is 0 < y < 
Similarly we may find the marginal f(x). We must restore the original function f(x,y)
f=subs(f,y,v);
f=subs(f,u,x);
fx=int(f,v,0,inf)
fx=subs(fx,CT,C);
pretty(fx)
fx =
CT*exp(-x)
exp(-x)
- ----------exp(-b) - 1
For values of 0< x < b.
The marginal density functions can be plotted over 0 < x < 1 and 0 < y <10. We assume that y= 10
is equivalent to y = We use the step function (In Maple the Heaviside function) to limit the range
of x in the plot
C0=double(subs(C,b,1))
C0 =
1.5820
xp=-.1:0.051:5;
fx=C0*exp(-x);
fxp=subs(fx,x,xp).*( mfun('Heaviside',xp) 1));
fx=C0*exp(-x);
fy=exp(-y)
mfun('Heaviside',xp-
fyp=subs(fy,y,xp).*mfun('Heaviside',xp);
plot(xp,fxp,xp,fyp)
fy =
exp(-y)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-1
0
1
2
3
4
5
The cumulative distribution function associated with the marginal function can be computed
FX=int(CT*exp(-u),u,0,x)
FX =
-CT*exp(-x)+CT
and similarly for y
FY=int(exp(-v),v,0,y)
FY =
-exp(-y)+1
The distribution can be plotted over their ranges respectively
C0=double(subs(C,b,1));
FX=subs(FX,CT,C0);
FXP=subs(FX,x,xp).*( mfun('Heaviside',xp) 1));
FY=1-exp(-y)
FYP=subs(FY,y,xp).*mfun('Heaviside',xp);
plot(xp,FXP,xp,FYP)
FY =
1-exp(-y)
mfun('Heaviside',xp-
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1

0
1
2
3
4
5
