Distribution of \(X\) | density function | \(\mathbb{E}[X]\) | \(\mathbb{V}[X]\) | parameter bounds | |
---|---|---|---|---|---|

Poisson()$ | \(e^{-\lambda}\lambda^x / x!\) | \(\lambda\) | \(\lambda\) | \(\lambda > 0\) |

Assume the population distribution of interest is \(\text{Poisson}(\lambda = 7)\).

Generate a random sample of \(N = 314\) observations from the population of interest and store it into a vector named

`x`

.Write in R a function that returns negative one times the simplified log-likelihood for

`N`

observations from the population of interest. The function signature should be`ll_poisson <- function(l, X)`

.Use the function

`optim(...)`

, along with the vector named`x`

that you generated in`a.`

, to find the maximum likelihood estimate of \(\lambda\), \(\hat{\lambda}\) by minimizing the function your wrote in part`b.`

. Remember that \(\lambda\) is bounded below by \(0\), such that you should set the`lower`

bound as an argument to`optim()`

.In a for loop of length

`R = 1001`

, repeatedly sample with uniform probabilities and with replacement observations from the vector`x`

. For each re-sampled vector, calculate and store your estimate, found in the element name`par`

returned from`optim()`

. Don’t forget to pre-allocate your memory.Make a density plot of the

`R`

sample statistics \(\hat{\lambda}\).Describe the sampling distribution of the statistic \(\hat{\lambda}\). What approximate shape does it take on? Why? Explain.

Will the width of the sampling distribution get wider or narrower as your sample size increase? If you’re not sure, change

`N`

above and repeat`a.`

through`e.`

to find out.Use the function

`quantile()`

, along with the vector of bootstrap re-sampled estimates of \(\hat{\lambda}\), to calculate an 90% confidence interval for population parameter \(\lambda\). Interpret this confidence interval.