Calculate a value for the integral $$ I=\int_{0}^{1} \frac{x^{-1 / 2}}{\mathrm{e}^{x}+1} \mathrm{~d} x, $$ using the importance sampling formula, Eq. (10.42), with \(w(x)=x^{-1 / 2}\), as follows. a) Show that the probability distribution \(p(x)\) from which the sample points should be drawn is given by $$ p(x)=\frac{1}{2 \sqrt{x}} $$ and derive a transformation formula for generating random numbers between zero and one from this distribution. b) Using your formula, sample \(N=1000000\) random points and hence evaluate the integral. You should get a value around \(0.84\).

Let's start by understanding the integral we need to calculate: \[ I=\int_{0}^{1} \frac{x^{-1 / 2}}{\textrm{e}^{x}+1} \textrm{~d} x \] Our goal is to evaluate this integral using importance sampling. In this method, a different, more convenient probability distribution is used, rather than directly evaluating the integral. By doing this, we can estimate the integral more efficiently.
First, we need to identify an appropriate weight function that will make our computations easier. Here, the given weight function is: \[ w(x)=x^{-1 / 2} \] Next, we need to determine the corresponding probability distribution from which we will draw our random samples. Understanding this step is crucial because it allows us to transform the initial integral into a more manageable form using the importance sampling technique.

Using the given weight function, we can derive the probability distribution: \[ p(x)=\frac{1}{2 \sqrt{x}} \] This function represents the probability of drawing a particular value of x when performing the sampling. It's essential to ensure that this distribution is normalized over the interval [0, 1].
To generate random samples based on this distribution, we need to find a transformation formula. This involves taking a uniformly distributed random variable and transforming it to follow our derived distribution: \[ p(x) = \frac{1}{2 \sqrt{x}} \] To find the transformation, we perform the following steps:

• Calculate the cumulative distribution function (CDF) of p(x).
• Invert the CDF to express x as a function of a uniformly distributed random variable u.

For our problem, the cumulative distribution function (CDF) is: \[ F(x) = \int_{0}^{x} p(t) \textrm{d}t = \int_{0}^{x} \frac{1}{2 \sqrt{t}} \textrm{d}t \] After performing the integration, we get: \[ F(x) = \sqrt{x} \] Inverting this CDF, we find:
\[ x = u^2 \] where u is a uniformly distributed random variable between 0 and 1. This transformation formula enables us to generate random numbers following the desired distribution.

Now, we have a method to generate random samples based on our derived probability distribution. Using the formula:

\[ x = u^2 \] we can generate the necessary random numbers and use these to estimate our integral through importance sampling. Here’s the process in simple steps:

• Generate N uniform random samples u, where N is large (e.g., 1000000).
• Transform these uniform samples using x = u^2 to follow the distribution p(x).
• Estimate the integral using the transformed samples.

For the estimation, if we denote our transformed samples as x_i for i = 1, 2, ..., N, then the integral can be approximated by:
\[ I \approx \frac{1}{N} \tilde{f}(x_i) \] where \[ \tilde{f}(x) = \frac{1}{\textrm{e}^{x} + 1} \] Each x_i is calculated based on the uniform samples using the transformation formula. By executing this sampling and computing the average value, we get the estimated integral. This should give us a value around 0.84 as expected.