Generating Discrete Random Variables

general method of generating discrete random variables of a particular discrete distribution

consider an arbitrary discrete random variable 𝐗 that takes values 𝑥₀, 𝑥₁, … with probabilities 𝑝₀, 𝑝₁, …

• 𝑝_i = 𝐏{𝐗 = 𝑥_i}
• ∑𝑝_i = 1

simulating 𝐗 can be done with the following steps

1. Divide the interval [0, 1] into subintervals as shown in Figure,
   1. 𝐴₀= [0, 𝑝₀)
   2. 𝐴₁= [𝑝₀, 𝑝₀+𝑝₁)
   3. 𝐴₂= [𝑝₀+𝑝₁, 𝑝₀+𝑝₁+𝑝₂)
   4. etc
2. subinterval 𝐴_iwill have length 𝑝_i; there may be a finite or infinite number of them, according to possible values of 𝐗
3. obtain a Standard Uniform random variable 𝐔 from a random number generator or a table of random numbers
4. if 𝐔 belongs to 𝐴_i, let 𝐗=x_i

from the Uniform Distribution, it follows that:

𝐏{𝐗 = 𝑥_i} = 𝐏{𝐔 ∈ 𝐴_i} = 𝑝_i

Hence, the generated variable 𝐗 has the desired distribution