Question

Consider a multinomial trial with outcomes \(1,2, \ldots, k\) and respective probabilities \(p_{1}, p_{2}, \ldots, p_{k} .\) Let ps denote the \(\mathrm{R}\) vector for \(\left(p_{1}, p_{2}, \ldots, p_{k}\right) .\) Then a single random trial of this multinomial is computed with the command multitrial (ps), where the required \(\mathrm{R}\) functions are: \({ }^{2}\) (a) Compute 10 random trials if \(\mathrm{ps}=\mathrm{c}(.3, .2, .2, .2, .1)\). (b) Compute 10,000 random trials for ps as in (a). Check to see how close the estimates of \(p_{i}\) are with \(p_{i}\).

Short answer

The commands to compute the random trials are trials_10 <- rmultinom(10, 1, c(.3, .2, .2, .2, .1)) and trials_10000 <- rmultinom(10000, 1, c(.3,.2,.2,.2,.1)) for 10 and 10,000 trials respectively. The command to estimate the probabilities is estimate <- colSums(trials_10000)/10000. The closer the estimated probabilities are to the original probabilities, the better the estimate.

Step-by-step solution

Generating Random Trials with Small Size

The task is to compute 10 random trials with ps = c(.3, .2, .2, .2, .1). In R, use the rmultinom function from the stats package. The rmultinom function generates multinomial random numbers. The syntax is rmultinom(n, size, prob), where n is the number of random numbers to generate, size is the number of trials, and prob is a vector of probability weights. For this task, the command will be trials_10 <- rmultinom(10, 1, c(.3, .2, .2, .2, .1)).

Generating Random Trials with Large Size

Similarly, generate 10,000 random trials with the same probabilities. The R command will be trials_10000 <- rmultinom(10000, 1, c(.3,.2,.2,.2,.1)).

Estimating Probabilities

The next step after generating these random trials is to compute the estimates of the probabilities p_i. This can be done by taking the sum of each outcome and dividing by the total number of trials. In R, if the output was stored in a matrix (where each row corresponds to an outcome and each column corresponds to a trial), the command to get the estimates will be estimate <- colSums(trials_10000)/10000.

Compare Estimated Probabilities with Original Probabilities

Upon obtaining the estimated probabilities, these should be compared with the original probabilities, i.e., c(.3, .2, .2, .2, .1). The comparison can be done visually by creating a table of the original probabilities and estimated probabilities, or numerically by calculating the absolute difference between the estimated and original probabilities. If the difference between the original and estimated probabilities is smaller, it can be inferred that the estimate is closer to the actual value.