44  Workshop Activities

Exercise 1: Estimating an Integral

Estimate the value of the following integral using Monte Carlo integration:

\[ I = \int_0^1 e^{-x^2}\,dx. \]

  1. Write down the Monte Carlo estimator for \(I\).
  2. Implement the estimator in R or Python, and compare your estimate to the true value of the integral (which can be computed using numerical integration).

Exercise 2: Estimating a Probability

Estimate the value of the following probability using Monte Carlo integration:

\[ P(X > 2.5), \quad X \sim N(0,1). \]

  1. Write down the Monte Carlo estimator for \(P(X > 2.5)\).
  2. Implement the estimator in R or Python, and compare your estimate to the true value of the probability (which can be computed using the cumulative distribution function of the normal distribution).

Exercise 3: Estimating the Area Under a Curve

Estimate the area under the curve:

\[ y = \sin(x), \quad 0 \le x \le \pi. \]

  1. Explain how to solve this analytically.
  2. Write down the Monte Carlo estimator for the area under the curve.
  3. Implement the estimator in R or Python, and compare your estimate to the true value of the area (which can be computed using numerical integration).

Exercise 4: Estimating an Expectation from an Exponential Distribution

Estimate \(\mathbb{E}[X^2]\) where \(X \sim \text{Exp}(\lambda = 1)\).

  1. Explain how to solve this analytically.
  2. Write down the Monte Carlo estimator for \(\mathbb{E}[X^2]\).
  3. Implement the estimator in R or Python, and compare your estimate to the true value of the expectation (which can be computed using the properties of the exponential distribution).

Exercise 5: High-Dimensional Integration

Estimate the value of the following integral using Monte Carlo integration:

\[ I = \int_{[0,1]^5} \exp\left(-\sum_{i=1}^5 x_i^2\right) dx_1 dx_2 dx_3 dx_4 dx_5. \]

This is a five-dimensional integral, which can be challenging to compute using traditional numerical integration methods. Use Monte Carlo integration to estimate the value of \(I\).

  1. Write down the Monte Carlo estimator for \(I\).
  2. Implement the estimator in R or Python, and compare your estimate to the true value of the integral (which can be computed using numerical integration or known properties of the multivariate normal distribution).

Exercise 6: Reliability Simulation

Suppose a machine fails if the stress S exceeds the strength R.

Let

\[ S \sim N(50, 10^2), \quad R \sim N(70, 15^2), \]

independetly. Estimate the probability that the machine fails, i.e., \(P(S > R)\), using Monte Carlo simulation.

  1. Write down the Monte Carlo estimator for \(P(S > R)\).
  2. Implement the estimator in R or Python, and compare your estimate to the true value of the probability (which can be computed using properties of the normal distribution).
  3. Discuss the advantages of using Monte Carlo simulation for reliability analysis in engineering.