Monte Carlo Practice Problems

1. Given a function, use Monte Carlo to determine the area under the curve in the region indicated:
   
   1. \(f(x)=3x+2\) for \(0\leq x\leq 3\)
   2. \(f(x)=x^2\) for \(-2\leq x\leq 2\)
   3. \(f(x)=ln(x+1)\) for \(0\leq x\leq 2\)
   4. \(f(x)=\exp(-x)\) for \(0\leq x\leq 2\)
   5. \(f(x)=\exp(-x^2)\) for \(-3\leq x\leq 3\),
   6. \(f(x)=\sqrt{x}\) for \(0\leq x\leq 3\)
   7. \(f(x)=\frac{1}{1+x^2}\) for \(-1\leq x\leq 1\)
   8. \(f(x)=\frac{1}{1+e^{-x}}\) for \(-1\leq x\leq 1\)
2. Estimate the area of an ellipse given by the equation $$\frac{x^2}{9}+\frac{y^2}{16}=1$$
3. Use Monte Carlo to estimate the mean of the X where \(X\sim |N(0,1)|\)
4. Use Monte Carlo to calculate \(\mathbf{E}[e^X]\) where \(X\sim N(0,1)\)
5. Use Monte Carlo to calculate \(\mathbf{E}[X^3]\) where \(X\sim N(0,1)\)
6. Use Monte Carlo to calculate \(\mathbf{E}[e^X]\) where \(X\sim Bin(n=25,p=0.75)\)
7. Use Monte Carlo to calculate \(\mathbf{E}[X^2]\) where \(X\sim Bin(n=14,p=0.4)\)
8. Use Monte Carlo to calculate \(\mathbf{E}[\frac{1}{1+x}]\) where \(X\sim Bin(n=20,p=0.5)\)
9. Let \(X\sim U(0,1)\) and \(x_1,x_2\) to be two realizations from that distribution. Use Monte Carlo to determine the probability that \(|x_1|+|x_2|\geq 1.25\)
10. Use Monte Carlo to estimate the volume equivalent of a 4-dimensional sphere with radius 1.