It is a shifted gamma.  (X-1) is a gamma distribution with parameters a and b, so generate from gamma(a,b) and add 1 to it.  To generate gamma several methods are available, you have R-program also.

Thanks for your answer. If Y=X-1has a gamma distribution with parameters a and b, then f(y)=const. ya-1e-by . By considering Z=Y+1, I have

f(z)=const. (z-1)a-1e-b(z-1). I think based your suggestion, I generate sample from f(z) that is not same with f(x). Please guide me.

hi

compute F(x) using pgamma function in R.

generate u by runif(1,0,1)

solve F(x)-u=0, using uniroot function.

check that x falls in the domain of density or not.

Yes, you are correct.  I am wrong. 

Dear Dr. Waghei,

Thanks for your answer. I can calculate F(t) as the integral of f(x) from 1 to t. Then I solve F(x)-u=0, where u is a random sample from U(0,1).

Dear professor Kundu,

The function f(x) is log-concave. Can I use Devroye Algorithm to generate sample from f(x)? Note that X is truncated. What do you think?

You may use Metropolis-Hasting (MH) method for sample generation. This is a particular method of MCMC, and you have no need to compute normalising constant in this manner. I hope this work excellent with your pdf. For details about MH, you also read the link given below.

dx.doi.org/10.1080/03610918.2015.1076469

http://www.r-bloggers.com/visualising-the-metropolis-hastings-algorithm/

Presentation Talk at the IISA conference 2015, Pune University,

https://darrenjw.wordpress.com/2012/06/04/metropolis-hastings-mcmc-when-the-proposal-and-target-have-differing-support/

please feel free to contact me.

https://theoreticalecology.wordpress.com/2010/09/17/metropolis-hastings-mcmc-in-r/

A roughly R code is written below, please write your code for truncated distribution by following the above link which one sent by me.

#samplee generation for your distribution with parameter

a=2; b=1

dist

Easily define a function as below, then generate a set of random numbers form a desired distribution within your variable(s) domain(s). later on, coerce the generated variables to the function.

1) define the function:

       f