Write a program in \(\mathrm{R}\) that first simulates 1000 observations from a PoIS\(\operatorname{SON}(10)\) distribution. For each observation, simulate a \(\operatorname{BinOmial}\left(n, \frac{1}{2}ight)\) observation where \(n\) is equal to the corresponding observation from the POIS\(\operatorname{SON}(10)\) distribution. Repeat this experiment five times and plot the resulting sequences of ratios of the BINOMIAL observations to the POISSON observations. Describe the plots and address whether the behavior in the plots appears to indicate that the theory given in Exercise 5 has been observed.

Exercise 5

Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) and \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be sequences of independent random variables. Suppose that \(Y_{n}\) is a \(\operatorname{POISSON}(\theta)\) random variable where \(\theta\) is a positive real number. Suppose further that, conditional on \(Y_{n}\), the random variable \(X_{n}\) has a \(\operatorname{Binomial}\left(Y_{n}, \tauight)\) distribution for all \(n \in \mathbb{N}\) where \(\tau\) is a fixed real number in the interval \([0,1]\). Prove that \(X_{n}=O_{p}\left(Y_{n}ight)\) as \(n ightarrow \infty\).