Question

Recall that \(\log 2=\int_{0}^{1} \frac{1}{x+1} d x\). Hence, by using a uniform(0,1) generator, approximate log \(2 .\) Obtain an error of estimation in terms of a large sample \(95 \%\) confidence interval. 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.

Short answer

The implemented R function performs multiple simulations to approximate \( \log 2 \) using a Monte Carlo method, and then calculates a 95% confidence interval to estimate the error. The approximate value, after 10,000 simulations, should be close to the true value of \( \log 2 \). The error estimation gives a range in which the true value exists with 95% confidence. This range can be used to verify how good the approximation is.

Step-by-step solution

Understanding Monte Carlo Method for Integration

Monte Carlo methods are often used to compute complex integrals. In this case, this method will be used to approximate the integral of \( \int_{0}^{1} \frac{1}{x+1} d x \) which equals to \( \log 2 \). The integral is pretty much the average of the function evaluated at random samples, so if 'n' random uniform values in [0,1] are denoted as \( X_{1}, X_{2}, ..., X_{n} \), the approximation of the integral, and thus \( \log 2 \), will be the average of \( \frac{1}{X_{i}+1} \).

Implementing R Function for Estimate and Error of Estimation

The next step is to implement an R function to generate 'n' uniform random numbers, calculate the approximation of log 2 and estimate the error. Below is an example of such a function. ```Rlog2_approx <- function(n) { X <- runif(n) estimations <- 1/(X+1) log2_est <- mean(estimations) error <- qnorm(0.975)*sd(estimations)/sqrt(n) return(list(log2_est, error))}```This function generates 'n' uniform random numbers between 0 and 1, computes the value of function \( \frac{1}{x+1} \) at each of these points, and calculates the average (log2_est). The error is then calculated as the standard deviation of these computations divided by square root of 'n', all multiplied by the 0.975-quantile of the standard normal distribution, corresponding to a 95% confidence interval.

Performing Simulations and Comparing to True Value

The function can now be used to perform 10,000 simulations and the result compared to the true value of \( \log 2 \). ```Rsimulation_result <- log2_approx(10000)print(simulation_result) ```This will print out the estimated value of \( \log 2 \) with a 95% confidence interval for the error of estimation. Compare the result to the true value gotten with the R function “log(2)”. You can use something like:```Rtrue_val <- log(2)print(true_val)```to output the true value.

Key concepts

Mathematical Statistics

Mathematical statistics is a branch of mathematics that focuses on the collection, analysis, interpretation, and presentation of data. It provides the theoretical foundation for making inferences from empirical information and plays a crucial role in many fields, from social sciences to medicine. One of its key techniques is the Monte Carlo method, which is a computational algorithm that relies on repeated random sampling to obtain numerical results. In the context of our exercise, the integral of the function representing the natural logarithm of 2 is approximated using this method.

The idea is to simulate a large number of values within a set range—in this case, between 0 and 1—based on the uniform distribution. By evaluating the integrand at these points and taking the average, one can approximate the value of the integral. It's essential to understand that the larger the sample size, or the number of simulations, the closer the approximation tends to be to the actual value. Furthermore, estimating the error and its associated confidence interval is integral in assessing the reliability of the simulation results.

R Programming

R is a powerful programming language and software environment for statistical computing and graphics. It is widely used among statisticians and data scientists for data analysis and developing statistical software. In the given exercise, R is the tool of choice to perform Monte Carlo integration. The language's capacity to handle complex computations with ease, as well as its extensive libraries for statistical analysis, makes it perfect for creating simulations like these.

Writing functions in R is straightforward, which simplifies the creation of customizable scripts for statistical procedures. The provided R function 'log2_approx' demonstrates how one can encapsulate the entire process of generating random variables, calculating the log approximation, and computing the error estimate all in a single, reusable function. This is a testament to R's efficiency and suitability for tasks involving repetitive calculations and statistical evaluations.

Confidence Interval Estimation

Confidence interval estimation is a fundamental concept in statistics for quantifying the uncertainty of an estimate. A confidence interval provides a range of values, derived from the data sample, that is likely to contain the value of an unknown population parameter. In practice, this means we can be, for example, 95% confident that the true value lies within our calculated range.

The establishment of a confidence interval involves determining the correct distribution for your data--in this case, the formula in the function involves the standard normal distribution, as we are dealing with a large sample size. The width of the interval depends on the standard deviation of the data and the sample size, captured within the formula for error with the 'qnorm' function in R, which gives the critical value from the normal distribution. By including such a confidence interval in our Monte Carlo simulation results, we can present not only an estimate but also the statistical certainty of that estimate.

Uniform Distribution

The uniform distribution is the underlying statistical theory that supports the Monte Carlo simulation used in our exercise. In a uniform distribution, every value between the lower and upper bounds of the distribution has equal probability of occurring. The 'runif' function in R generates random numbers following a uniform distribution, which are crucial for the Monte Carlo method used to approximate the logarithm of 2.

In the context of uniform distribution, we assume a continuous range between 0 and 1 where any point has an equal chance of being selected. This equal probability is important for the random sampling aspect of the Monte Carlo integration, ensuring that every possible sample point contributes to the approximation without bias. This characteristic leads to a fair representation of the function across its entire domain when approximating the integral.