Question

Give an efficient algorithm to simulate the value of a random variable with probability mass function

p1 = .15 p2 = .2 p3 = .35 p4 = .30

Short answer

The algorithm is define $X$using the Uniform continuous distribution $U$over $(0,1)$.

Step-by-step solution

Given Information

We have given the probability mass function

$p_1=.15p_2=.2p_3=.35p_4=.30$

Simplify

Generating $U~Unif(0,1)$.

Now, declare $X$as follows

$X=\left\{\begin{array}{ll}1,& U\in \left(0,0.15\right)\\ 2,& U\in \left(0.15,0.35\right)\\ 3,& U\in \left(0.35,0.7\right)\\ 4,& U\in \left(0.7,1\right)\end{array}\right.$

Consider that because of the construction we have, for example

$P(X-1)=P(U\in (0,0.15\left)\right)=0.15=p_1$

and similarly for all other values

$P(X=i)=p_i$

So, we have$X$follows the required discrete distribution.