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I am solving optimization problems where I am trying to find the minimum of a function over some sample space $\mathcal{X}$, i.e., $\min\,f(x):x\in\mathcal{X}$. Now the optimization algorithm I am using is based on trial points $x'$ which are sampled from $\mathcal{X}$. For sake of argument, let's say $\mathcal{X}\in[0,1]$ is the unit interval.

Now I have been solving some problems where the solution lies along the boundary, i.e., $x=0$ or $x=1$ could be the solution to the minimization problem. Now, the way I have been picking my potential solutions (trial points) is to sample $x'$ from a Uniform(0,1) distribution.

No, what my question really is, is whether or not I will ever sample 0 or 1 from that Uniform distribution. From a practical point of view I don't think it will occur, however, from a theoretical point of view I am also not sure. Because isn't the probability of sample any one single number from a continuous distribution equal to exactly 0? Or is there some positive probability that I will sample the endpoints of the interval?

However, running some R code sampling from a Uniform(0,1) distribution 100,000,000 times I am able to sample 1, but not 0 (well maybe in machine precision it is?)

> x = runif(100000000) > min(x) [1] 2.142042e-08 > max(x) [1] 1 >
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    $\begingroup$ The definition of "continuous distribution" is that there will not be positive probability of sampling any given value. You can even specify certain infinite sets of values and they will still have zero probability. BTW, what is the output of 1-max(x)? $\endgroup$ Commented Aug 17, 2016 at 17:24
  • $\begingroup$ You're right, it is non-zero. $\endgroup$ Commented Aug 17, 2016 at 17:28

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When using R's runif by design you won't be able to sample neither 0, nor 1. Check it's source code:

double u; /* This is true of all builtin generators, but protect against user-supplied ones */ do {u = unif_rand();} while (u <= 0 || u >= 1); return a + (b - a) * u;
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  • $\begingroup$ That is not true though. Above in my question I sampled 100,000,000 uniform random variables and I do sample 1 (you can see that max(x)=1). $\endgroup$ Commented Aug 17, 2016 at 17:12
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    $\begingroup$ @RustyStatistician I doubt it. That's probably a matter of how to represent the float to your console. $\endgroup$ Commented Aug 17, 2016 at 17:21

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