Question

Approximate log 2 by using a uniform \((0,1)\) generator. Obtain an error of estimation in terms of a large sample \(95 \%\) confidence interval. If you have access to the statistical package \(\mathrm{R}\), write an \(\mathrm{R}\) function for the estimate and the error of estimation. Obtain your estimate for 10,000 simulations and compare it to the true value. Hint: Recall that \(\log 2=\int_{0}^{1} \frac{1}{x+1} d x\).

Short answer

The log 2 was approximated using a uniform (0,1) generator via a Monte Carlo integration by writing an R function. Then the error of estimation was obtained by simulating that function for 10,000 times and calculating the standard error. Finally, the estimated value was compared with the actual value of log 2.

Step-by-step solution

Understanding the Concept

Note that the integral of 1/(x+1) from 0 to 1 equals to log 2. Therefore, the task is to estimate this integral using Monte Carlo method. Monte Carlo method involves repeated sampling to obtain numerical results. In this exercise, Monte Carlo method can be applied for approximating the integral by drawing random uniform numbers, applying the function 1/(x+1) on them and calculating the mean.

Writing the R Function

Write a function in R to simulate this process. First, make a function that generates n uniform random numbers, applies the function 1/(x+1) to them and calculates the mean. Then, make another function to simulate this process several times (for example, 10,000 times as in the exercise) and calculates the mean and standard error. The standard error serves as an estimate for the error of estimation in this context.

Calculate the Confidence Interval

Next, calculate the 95% confidence interval for your estimation. In R, this can be done using the formula for a confidence interval, which is mean ± (1.96 * standard error(s)). 1.96 is the approximate value of the 97.5 percentile point of the normal distribution used in 95% confidence intervals.

Compare with the True Value

Lastly, compare the estimate acquired from the simulation to the true value. The true value of log 2 can be calculated in R using the log function: log(2).

Key concepts

Confidence Interval

Understanding the concept of a confidence interval is crucial when dealing with statistical estimation. A confidence interval gives us a range within which we expect the true parameter of a population to fall a certain percentage of the time. For instance, a 95% confidence interval implies that if we were to take 100 different samples and calculate 100 different confidence intervals, we would expect about 95 of them to contain the true parameter value.

When they're constructed using the mean and standard error as in our Monte Carlo simulation exercise, the interval is typically symmetrical around the sample mean. It is calculated by taking the sample mean and adding and subtracting the product of a critical value (derived from the desired level of confidence) and the standard error. In the exercise, the critical value for a 95% confidence interval is 1.96, corresponding to the 97.5th percentile of a standard normal distribution. The calculated interval then provides a practical indication of the uncertainty around our estimate.

R Programming

R is a powerful programming language used extensively in data analysis, statistics, and graphical representations. In the context of our Monte Carlo simulation exercise, R is ideal due to its rich set of packages and functions that can handle statistical operations efficiently. Writing a function in R, as suggested in Step 2 of our exercise, involves creating a script that can generate random numbers, apply the function to each number, and compute statistical estimates like the mean and standard error.

R's ability to perform simulations repeatedly through looping or vectorized operations is central to completing the exercise effectively. Moreover, R’s built-in functions for calculating logarithms and performing basic arithmetic make it a convenient tool for comparing the simulated estimate of \(\log 2\) with its true value, enhancing the learning outcome for students exploring numerical integration via Monte Carlo simulation.

Numerical Integration

Numerical integration is a key area of computational mathematics. It comes into play when finding exact solutions is problematic, which is often the case with complex integrals. The Monte Carlo method, as seen in our exercise, is a stochastic approach to numerical integration that relies on the law of large numbers. By uniformly sampling random points within the domain and calculating the average of a function at these points, we can approximate the area under the curve, which corresponds to the integral's value.

This method is very versatile and can be used for high-dimensional integrals where traditional techniques falter. However, it's important to note that the accuracy of Monte Carlo integration improves with the number of samples, which is why large numbers of simulations, such as the 10,000 iterations in our example, are often used to achieve a more accurate estimation of integrals like \(\log 2\).

Statistical Estimation

Statistical estimation involves using sample data to estimate population parameters. One common estimator is the sample mean, which serves as an estimation of the population mean. In our Monte Carlo simulation for approximating \(\log 2\), we treat the mean of our function values at randomly generated points as the estimator for our integral.

Two essential components of statistical estimation are the point estimate and the error of estimation. The point estimate provides a single best guess, while the error of estimation reflects the uncertainty associated with this estimate. In Monte Carlo simulations, the standard error, which we calculate as part of our procedure, provides a measure of this uncertainty. By understanding these principles, students can better grasp the rationale behind the steps of the given exercise and the significance of the confidence interval as a measure of estimation accuracy.