Question

Suppose it is relatively easy to simulate from the distributions \(F_{i}, i=1,2, \ldots, n .\) If \(n\) is small, how can we simulate from $$ F(x)=\sum_{i=1}^{n} P_{i} F_{i}(x), \quad P_{i} \geqslant 0, \quad \sum_{i} P_{i}=1 ? $$ Give a method for simulating from $$ F(x)=\left\\{\begin{array}{ll} \frac{1-e^{-2 x}+2 x}{3}, & 0

Short answer

To simulate from the given distribution \(F(x)\), first identify the component distributions \(F_1(x)\), \(F_2(x)\) and their corresponding weights \(P_1\), \(P_2\). Then, generate a random number \(p\) uniformly between 0 and 1. Based on \(p\), choose the appropriate component distribution and generate a random sample from it using its PDF. Record the sample as a simulated value from the overall distribution F(x).

Step-by-step solution

Create Method for Simulating from Weighted Sum of Distributions

To simulate from a distribution \(F(x) = \sum_{i=1}^{n} P_{i} F_{i}(x)\), where \(P_i \geq 0\) and \(\sum_i P_i = 1\), you can follow these steps: 1. Define and normalize the weights \(P_i\). 2. Generate a random number, say \(p\), uniformly between 0 and 1. 3. Determine which component distribution \(F_i(x)\) should be used based on the random number \(p\) in relation to the corresponding weights \(P_i\). 4. Generate a random sample from the chosen distribution \(F_i(x)\). 5. Record the sample as a simulated value from the overall distribution F(x).

Identify Component Distributions and Weights

For the distribution: $$ F(x)=\left\\{\begin{array}{ll} \frac{1-e^{-2 x}+2 x}{3}, & 0<x<1 \\\ \frac{3-e^{-2 x}}{3}, & 1<x<\infty \end{array}\right. $$ We can treat it as a mixture of two component distributions, \(F_1(x)\) and \(F_2(x)\), with weights \(P_1\) and \(P_2\) such that \(\sum_i P_i = 1\). We have: - \(F_1(x) = (1-e^{-2 x}+2 x)\) for \(0 < x < 1\). - \(F_2(x) = (3-e^{-2 x})\) for \(1 < x < \infty\). - \(P_1 = \frac{1}{3}\) and \(P_2 = 1 - P_1 = \frac{2}{3}\).

Simulate from the Mixture Distribution

Now, we would like to simulate from the given distribution \(F(x)\). We use the method from Step 1 with the identified component distributions and weights. Follow these steps: 1. Normalize the weights: \(P_1 = \frac{1}{3}\) and \(P_2 = \frac{2}{3}\). 2. Generate a random number \(p\) uniformly between 0 and 1. 3. Determine which component distribution to use based on \(p\): - If \(0 \leq p < \frac{1}{3}\), choose \(F_1(x)\). - If \(\frac{1}{3} \leq p < 1\), choose \(F_2(x)\) (1 ≤ x < ∞). 4. Generate a random sample from the chosen distribution: - If \(F_1(x)\) is chosen, simulate a value from its PDF, which is given by \(f_1(x) = \frac{d (1-e^{-2 x}+2 x)}{dx} = 2 - 2e^{-2x}\), using an appropriate method (e.g., Inverse Transform Sampling or Rejection Sampling). - If \(F_2(x)\) is chosen, simulate a value from its PDF, which is given by \(f_2(x) = \frac{d (3-e^{-2 x})}{dx} = 2e^{-2x}\), using an appropriate method (e.g., Inverse Transform Sampling or Rejection Sampling). 5. Record the sample as a simulated value from the overall distribution F(x). This method allows you to simulate from the given distribution F(x). Repeat the process for multiple samples to analyze the properties of the distribution.