@Carl Love 

You are right! The sum must be one.

But, how can I do that?

@Carl Love 

Many thanks.

I noticed that if d is not equal to zero, the function performs better. This is why I want more simulation study on it.

Besides, I could not get closed form of the moments(Mean, Variance...) when d is not eqaul to zero;

Plesae, help even if it is with appoximate mement!

@vv 

Thanks for the hint.

But, if d is not equal to 0 , please, how do I generate descret random numbers from the fucntion to enable my study the parameetrs(r, b, d).

@Carl Love 

Ok! I shall be expecting you.

@Carl Love 

When I run the below codes, here is the error.

Error, (in Statistics:-Sample) unable to construct the random number generation parameters for _R, try to specify the initial range

restart;

B := proc (x) options operator, arrow; binomial(x+r-1, x)*((d*x+1)/(1+d*x+(1/2)*b))^x*((1/2)*b/(1+d*x+(1/2)*b))^r/(d*x+1) end proc;

B1 := Statistics:-Distribution(

Type = discrete,

ProbabilityFunction = B,

Support = 0 .. infinity,

DiscreteValueMap = (proc (n) options operator, arrow; infinity end proc));

r, b, d := 0.2e-1, 0.1e-1, 0.9e-3;

Statistics:-Sample(B1, 10, method = [discrete, range = 0 .. infinity])

@Carl Love 

Thanks.

Let me try it on my own function. 

@vv 

Yes! you are write it is not defined for arbitray value of (r,b,d). 

Please, how do I study the limiting values of each parameter?

I equally noticed that, but dont know how to study it in general.

Thanks. 

@Carl Love 

Really! But, I have been answering your questions promptly. 

The distribution is Discrete; x is a vereiable that assumes values from [0, infinity);

b,d, and r are parameters defined over real numbers.

How can I use the command(Statistics:-Sample) you are suggesting?

Many thanks 

@Markiyan Hirnyk 

Thank so much.

How can I represent the inverse analyticaly such that it will enable me carry out simulation study? 

@Carl Love 

Yes! it is a Probability Density Function but, not a standard one. Please, how can I do simulation study around the function?

@Carl Love 

Many thanks.

I actually want to run Stochastic Simulation to generate simulated data using the function. Thus, I thought of inverse function method.

I will appreciate if I can see other possible method to carry out simulation study on that fucntion.