 # I have generated random sample from uniform distribution (sample size n) and then using the inverse function method generated the exponential distribution. I have run this simulation N1, N2, and N3 times where N1 < N2 < N3. acapotivigl 2022-09-14 Answered
monte carlo simulation - confidence intervals construction
I am starting with Monte Carlo Simulation. I have run simulation to estimate the mean and the variance of the exponential distribution.
Simulation:
I have generated random sample from uniform distribution (sample size n) and then using the inverse function method generated the exponential distribution. I have run this simulation N1, N2, and N3 times where $N1.
Results:
All worked well, when I plot the results I can see that the distribution of means and variances tends to the normal distribution as the number of simulation increases.
Questions:
1) Would you please help and clarify how the empirical and theoretical confidence intervals should be derived?
2) what is the manifestation of the Law of Large Numbers in this exercise?
3) and what is manifestation of Central Limit Theorem in this exercise?
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Step 1
1. I don't know what you mean by a theoretical confidence interval, perhaps a confidence interval where you use the value of $\sigma$ which is known in advance? As for the empirical confidence interval, that would be
$\left(\overline{X}-\frac{{t}_{N,\alpha }S}{\sqrt{N}},\overline{X}+\frac{{t}_{N,\alpha }S}{\sqrt{N}}\right)$
where $\overline{X}$ is the sample mean, S is the sample standard deviation, N is the sample size, and ${t}_{N,\alpha }$ comes from Student's t distribution. Specifically, if you want your interval to have confidence $0<\alpha <1$, then ${t}_{N,\alpha }$ is the number b such that the probability for Student's t distribution with $N-1$ degrees of freedom to be between −b and b is $\alpha$. For large N this is well-approximated by the corresponding value from the standard normal distribution.
Step 2
2. You should see that your sample means are all close to the true mean if N is large. (In this form I am using the strong law of large numbers; you need a whole bunch of sample means to make sense of the weak law of large numbers).
Step 3
3. If you take a bunch of sample means for a large but fixed N, $\mu$ is the true mean and $\sigma$ is the true standard deviation, then you have a bunch of instances of $\overline{X}$. If you make a histogram of the quantity $\sqrt{N}\frac{\overline{X}-\mu }{\sigma }$ using these values of $\overline{X}$, the distribution you get should look approximately like a standard normal.

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