Question

Set up the alias method for simulating from a binomial random variable with parameters \(n=6, p=0.4\)

Short answer

Set up the alias method for simulating a binomial random variable with parameters \(n = 6\) and \(p = 0.4\): 1. Calculate binomial probabilities using the PMF formula: \(P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}\). 2. Set up an alias table of length \(n + 1 = 7\), with two arrays for probabilities and aliases. 3. Normalize probabilities by a factor of \(M = n + 1\). 4. Split probabilities into two groups: underfull (< 1) and overfull (> 1), and transfer probability mass between them until all bins are full. 5. Use alias sampling by generating a random index and uniform random number to simulate the binomial random variable.

Step-by-step solution

Calculate Binomial Probabilities

First, we will calculate the probability mass function (PMF) of the binomial distribution for the given parameters (\(n = 6, p = 0.4\)). The probability for each outcome (\(x\)) can be calculated using the formula: \[ P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} \] where \(P(X = x)\) is the probability of getting \(x\) successes, and \(\binom{n}{x}\) is the binomial coefficient, computed as: \[ \binom{n}{x} = \frac{n!}{x!(n - x)!} \] Compute probabilities for \(x = 0, 1, 2, \dots, n\).

Construct Alias Table

Once we have the probabilities for each outcome, we can set up the alias table. The alias table consists of two arrays: one for probabilities and one for aliases. The length of these arrays is equal to the number of possible outcomes, which is \(n + 1 = 7\). 1. Initialize the alias array with -1 and the probability array with the calculated probabilities. 2. Normalize each probability by a factor of \(M = n+1\), and represent the probabilities as \(p_iM\). 3. Split the probabilities into two groups: those smaller than 1 (underfull), and those greater than 1 (overfull). 4. For each underfull bin, find an overfull bin and transfer a portion of probability mass from the overfull bin to the underfull bin to make it full. 5. Update the alias array with the index of the overfull bin from which probability mass was transferred. The construction process continues until there are no underfull bins left.

Alias Sampling

After the alias table is set up, we can generate a binomial random variable in two simple steps: 1. Generate a random index (\(I\)) from 0 to \(n\) with equal probability. 2. Generate a uniform random number (\(U\)) between 0 and 1. 3. If \(U\) is less than the probability in the probability array at index \(I\), return \(I\); otherwise, return the alias array value at index \(I\). The value returned is the simulated binomial random variable.