# Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.)

Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal distributions. In this case, I would like to know,

1. Are there differences between

(i) when each item is sampled for, say 1000 times, and we choose the lowest value among 10 items.

And

(ii) when each item is sampled for once, and we choose the lowest value among 10 items. Then, we average these values over 1000 trials.

2. If there are differences, do each of the above have special meaning in probability study?

Any help would be appreciated.

• The description of procedure (i) is not clear, I guess you average each of the 1000 trials for each variable ("item") and then choose the lowest value. If this is so, then (i) and (ii) provide different quantities. Operation (i) will give you (an estimate of) $\min_i\{\mathbb{E}(x_i)\}$ whereas operation (ii) will give you (an estimate of) $\mathbb{E}(\min_i\{x_i\})$ – Stelios Aug 16 '14 at 7:47
• Yes, that is what I meant. Would you say one of them is more suitable for Monte-Carlo-Simulation method? (or it depends?) – Yui Park Aug 16 '14 at 15:05
• This is not a issue of "simulation suitability", rather, its about what you are looking to estimate with the help of simulation. These operations provide estimates of different things (they are not equivalent) – Stelios Aug 16 '14 at 15:09
• Okay, that's what I expected. I guess, min{E(x)} can simply be calculated. However, would there be a way to guess, E(min{x})? – Yui Park Aug 16 '14 at 15:18
• In some cases $\mathbb{E}(\min_i\{x_i\})$ can be computed in closed form. See here for such an example. – Stelios Aug 16 '14 at 16:23