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Chapter 10: Problem 11

Use the rejection method with \(g(x)=1,0

Short Answer

Expert verified
To generate random variates from the density function \(f(x) = 60x^3(1-x)^2\), we use the rejection method with \(g(x) = 1\) for \(0 < x < 1\). First, find the largest possible value of \(M\) such that \(f(x) \le Mg(x)\), which is \(M = 15\). Then, follow the algorithm: 1. Generate \(U\) and \(V\) from the uniform distribution on \((0,1)\). 2. Compute the ratio \(R = \frac{f(U)}{Mg(U)} = 4U^3(1-U)^2\). 3. Accept \(U\) as a random variate from \(f(x)\) if \(V \le R\), else repeat. This algorithm will generate random variates from the desired density function \(f(x)\) using the rejection method and the instrumental function \(g(x)\).

Step by step solution

01

Find the largest possible value of \(M\)

We will begin by finding the largest possible value of \(M\) such that \(f(x) \le M g(x)\) for all \(x\) in the interval \(0 < x < 1\). We will then derive the appropriate inequality: $$M g(x) \ge f(x)$$ $$M \ge \frac{f(x)}{g(x)}$$ Since \(g(x) = 1\) for \(0 < x < 1\): $$M \ge 60x^{3}(1-x)^{2}$$ We want to find the global maximum of the function \(h(x) = 60x^{3}(1-x)^{2}\) on the interval \([0,1]\). To do this, let's first find the critical points of \(h(x)\) by taking its derivative and setting it to 0: $$\frac{dh(x)}{dx} = 180x^{2}(-2x^2 + 3x -1) = 0$$ The critical points of \(h(x)\) are the solutions of the equation \(-2x^2 + 3x -1 = 0\). We can solve this quadratic equation using the quadratic formula: $$x = \frac{-3 \pm \sqrt{(-3)^2 - 4(-2)(-1)}}{2(-2)} = \frac{3 \pm \sqrt{1}}{4}$$ This yields the critical points \(x = \frac{1}{2}\) and \(x = 1\). Since \(h(x) = 0\) at \(x = 1\), the global maximum of \(h(x)\) occurs at \(x = \frac{1}{2}\). Plugging this value back into \(h(x)\), we get: $$M = h\left(\frac{1}{2}\right) = 60\left(\frac{1}{2}\right)^3 \left(1 - \frac{1}{2}\right)^2 = 15$$ We have found the largest possible value of \(M\) to satisfy the inequality \(f(x) \le M g(x)\), which is \(M = 15\).
02

Rejection method algorithm

Now that we have found the constant \(M\), we will describe an algorithm to generate random variates from \(f(x)\) using the rejection method and the given instrumental function \(g(x)\), which has a density \(g(x) = 1\) for \(0 < x < 1\). Note that \(g(x)\) is the uniform distribution over the interval \((0, 1)\). Here are the steps of the algorithm: 1. Generate a random variable \(U\) from the uniform distribution on the interval \((0, 1)\). 2. Generate another random variable \(V\) from the uniform distribution on the interval \((0, 1)\). 3. Compute the ratio \(R = \frac{f(U)}{Mg(U)} = \frac{60U^3(1-U)^2}{15} = 4U^3(1-U)^2\). 4. Accept \(U\) as a random variate from \(f(x)\) if \(V \le R\), else go back to step 1 and repeat. This algorithm will generate random variates from the desired density function \(f(x)\) using the rejection method and the given instrumental function \(g(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simulating Random Variables
Simulating random variables is a key concept in probability and statistical computing. This involves creating a model of random processes to imitate or reflect the behavior of a real-life scenario. In this context, simulation helps us understand and predict the behavior of different variables based on their probability distributions.
The cornerstone of simulating random variables is the generation of random numbers, which mimic how actual data might behave under similar conditions. For instance, to simulate a random variable that follows a specific distribution, programmers utilize algorithms that imitate the statistical properties of the distribution. The rejection method is an example of such an algorithm. It is particularly handy when simulating continuous random variables with complex density functions, where direct sampling isn't feasible.
By employing concepts like the rejection method, we can closely replicate the randomness inherent in natural processes, which is valuable in fields like finance, physics, and computer science.
Density Function
The density function is a fundamental concept in probability theory. It describes how the probability density of a continuous random variable is distributed over all possible values. For a random variable with a density function, the probability that the variable takes on a specific value is essentially zero. However, the density function allows us to calculate probabilities over intervals.
In mathematical terms, a density function is represented as a function \( f(x) \) that is always non-negative, and the integral over the entire space is equal to 1. This property ensures that the total probability across all values remains a hundred percent.
The exercise at hand involves the density function \(f(x) = 60x^{3}(1-x)^{2}\) for \(0 < x < 1\), which means the random variable closely follows this probabilistic pattern. Understanding the shape and range of this function is crucial for successfully implementing simulation methods like the rejection method.
Uniform Distribution
Uniform distribution is a simple yet fundamental concept in probability. It refers to a distribution where every event is equally likely to happen. The graph of a uniform distribution is flat, indicating equal probability across the range of possible outcomes.
In this exercise, the instrumental function \(g(x) = 1\) represents a uniform distribution over the interval \(0 < x < 1\). This uniform distribution becomes pivotal in the rejection method because we sample from this flat distribution and subsequently determine whether these samples are accepted or rejected.
Utilizing a uniform distribution simplifies the sampling process, as it provides a constant probability foundation to integrate more complex density functions. This makes it an effective groundwork for building and applying algorithms aimed at simulating more intricate probability distributions.
Algorithm Steps
Understanding the step-by-step process of any algorithm is crucial for successfully implementing it. The rejection method algorithm, as used in the exercise, consists of a few straightforward steps designed to generate random variables with a given density function.
  • First, draw a random variable \(U\) from the uniform distribution over the interval \(0, 1\), which serves as a candidate for our target distribution.
  • Next, draw another random variable \(V\) from the same uniform distribution to serve as a threshold indicator.
  • Compute the ratio \(R = \frac{f(U)}{Mg(U)}\), derived from the target density function \(f(x)\) and the uniform density function \(g(x)\).
  • Accept \(U\) if \(V \le R\); otherwise, discard \(U\) and repeat the process.
This methodological approach is both systematic and iterative, allowing for the generation of samples that adhere to the desired density function. The algorithm ensures that the samples accurately reflect the underlying probability distribution, capturing the essence of the function \(f(x)\) through repetitive trials and refinements.

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Most popular questions from this chapter

Explain how you could use random numbers to approximate \(\int_{0}^{1} k(x) d x,\) where \(k(x)\) is an arbitrary function. Hint: If \(U\) is uniform on \((0,1),\) what is \(E[k(U)] ?\)

The following algorithm will generate a random permutation of the elements \(1,2, \ldots, n .\) It is somewhat faster than the one presented in Example 1 a but is such that no position is fixed until the algorithm ends. In this algorithm, \(P(i)\) can be interpreted as the element in position \(i\) Step 1. Set \(k=1\) Step \(2 .\) Set \(P(1)=1\) Step \(3 .\) If \(k=n,\) stop. Otherwise, let \(k=k+1\) Step 4. Generate a random number \(U\) and let $$\begin{aligned} P(k) &=P([k U]+1) \\ P([k U]+1) &=k \end{aligned}$$ Go to step 3 (a) Explain in words what the algorithm is doing. (b) Show that at iteration \(k\) - that is, when the value of \(P(k)\) is initially set \(-P(1), P(2), \ldots, P(k)\) is a random permutation of \(1,2, \ldots, k\) Hint: Use induction and argue that $$\begin{array}{l} P_{k}\left\\{i_{1}, i_{2}, \ldots, i_{j-1}, k, i_{j}, \ldots, i_{k-2}, i\right\\} \\ \quad=P_{k-1}\left\\{i_{1}, i_{2}, \ldots, i_{j-1}, i, i_{j}, \ldots, i_{k-2}\right\\} \frac{1}{k} \\ \quad=\frac{1}{k !} \text { by the induction hypothesis } \end{array}$$

Let \(X\) be a random variable on (0,1) whose density is \(f(x) .\) Show that we can estimate \(\int_{0}^{1} g(x) d x\) by simulating \(X\) and then taking \(g(X) / f(X)\) as our estimate. This method, called importance sampling, tries to choose \(f\) similar in shape to \(g,\) so that \(g(X) / f(X)\) has a small variance.

Suppose it is relatively easy to simulate from \(F_{i}\) for each \(i=1, \ldots, n .\) How can we simulate from $$\text { (a) } F(x)=\prod_{i=1}^{n} F_{i}(x) ?$$ $$\text { (b) } F(x)=1-\prod_{i=1}^{n}\left[1-F_{i}(x)\right] ?$$

In Example \(2 \mathrm{c}\) we simulated the absolute value of a unit normal by using the rejection procedure on exponential random variables with rate \(1 .\) This raises the question of whether we could obtain a more efficient algorithm by using a different exponential density - that is, we could use the density \(g(x)=\lambda e^{-\lambda x} .\) Show that the mean number of iterations needed in the rejection scheme is minimized when \(\lambda=1\)
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