Regarding the PDF and CDF of two gamma distributed random varaibles!!

If X ~ gamma (m, p) with a shape parameter m and a scale parameter p and Y~ gamma (m, q) with a shape parameter m and a scale parameter q when X and Y are independent random variables then What will be the PDF and CDF of X+Y? How can I solve it in MatLab

Thanks in advance for help

Answer 1

[Torsten]

The density function for X+Y is given by

f_X+Y(z)=exp(-q*z)*p^(2*m)/(gamma(m))^2*integral_{x=0}^{x=z}(x*(z-x))^(m-1)*exp(-(p-q)*x) dx

Use MATLAB's "integral" to evaluate the integral for different values of z.

Best wishes

Torsten.

[kader]

Thanks for the answer. Is this pdf or CDF (Specially I need CDF)? Is it the "int" or "integral" function in the matlab? Can you please tell me MatLab code for example, m=1, p=1, q=1, z=0.2, using symbolic variable.

Sorry for your inconvenience.

Thanks in advance again

[Torsten]

Does this work ?

m=1;
p=1;
q=1;
z1=0.2;
z2=3;
% Compute PDF at z1
fun1=@(x)(x.*(z1-x)).^(m-1).*exp(-(p-q)*x);
PDF = exp(-q*z1)*p^(2*m)/gamma(m)^2*integral(fun1,0,z1);
% Compute CDF at z2
fun2=@(x,z)exp(-q*z).*(x.*(z-x)).^(m-1).*exp(-(p-q)*x);
CDF = p^(2*m)/gamma(m)^2*integral2(fun2,0,@(z)z,0,z2);

Best wishes

Torsten.

[kader]

Thanks for your answer. For pdf, I am getting a numerical value. But for CDF there is an error like as

[Torsten]

Try

fun2=@(z,x)exp(-q*z).*(x.*(z-x)).^(m-1).*exp(-(p-q)*x);
CDF = p^(2*m)/gamma(m)^2*integral2(fun2,0,z2,0,@(z)z);

Best wishes

Torsten.

[kader]

Thanks a lot. I am really grateful to you

[kader]

I am really grateful to you for my previous answers. Can you please help me one more time?

If Xi (i=0,1..) ~ gamma (m, p) with a shape parameter m and a scale parameter p and X are independent random variables then What will be the PDF and CDF of sum(X)? How can I solve it in MatLab? i.e, PDF and CDF of the sum of two gamma random variables with same shape and scale parameter. You can add this one with the previous answers.

[Torsten]

Little correction to my code:

m=1;
p=1;
q=1;
z1=0.2;
z2=3;
% Compute PDF at z1
fun1=@(x)(x.*(z1-x)).^(m-1).*exp(-(p-q)*x);
PDF = exp(-q*z1)*(p*q)^m/(gamma(m))^2*integral(fun1,0,z1);
% Compute CDF at z2
fun2=@(z,x)exp(-q*z).*(x.*(z-x)).^(m-1).*exp(-(p-q)*x);
CDF = (p*q)^m/(gamma(m))^2*integral2(fun2,0,z2,0,@(z)z);

Best wishes

Torsten.

[Torsten]

All Xi have the same scale parameter ?

Then the result under

http://math.stackexchange.com/questions/250059/sum-of-independent-gamma-distributions-is-a-gamma-distribution

applies.

Best wishes

Torsten.

[kader]

But for me X1=gamma(a,m) and X2=gamma(a,m), then what will be the pdf and cdf of

[Torsten]

X0+X1 ~ gamma(2*a,m)

In general:

X0+X1+...+Xn ~ Gamma((n+1)*a,m)

To calculate pdf and cdf, use gampdf and gamcdf.

Best wishes

Torsten.

Answer 2

[the cyclist]

This is a math or stats question, not a MATLAB question. But, you made me curious.

The answer is not trivial. This paper gives a solution. This post on another forum gives an explanation (and references the paper I just mentioned).

[kader]

Specially,I want to know how to solve in Matlab