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I am estimating $\theta$, where $\theta=\int_0^1 f(x)dx, \ \text{where}\ f(x)=x*e^{x/2}$
From a unifrm distibution I generate 1000 random numberers from 0 to 1, $x_1,x_2,...x_{1000},$.
$\hat{\theta}= 1/1000 * \sum_{i=1}^{1000}f(x_i) $
Henc, I am taking the average.
How do I construct a Confidence Interval for $\hat{\theta}$? Say 95%.
As far as I can read from Wikipedia, I think my method is called Monte Carlo integration but I am not sure.
Expanding on whuber's comment.
The way you make a confidence interval in this case is the same as any.
You have $X_1, X_2, \dots, X_n$ from a Uniform$(0,1$). You have a function $y = g(x) = xe^{x/2}$, so that $Y_1, \dots, Y_n$ are a random sample from a distribution with mean $\theta$ and some unknown variance $\sigma^2$. Once you take the Monte Carlo average, the central limit theorem kicks in and in order to make confidence intervals you just have to obtain an estimate of $\sigma^2$.
