A statistics students suggests using a binomial model for the number of games out of these ten that the Packers will win. Provide one or more reasons for why this model may not meet all of the assumptions for a binomial model. > [Write your answer here] ###7 Imagine this two-stage process to create a random variable $X$. In the first step, select a coin at random from a large bag of coins where each coin as a different head probability, $P$. The distribution of $P$ is uniform between 0 and 1, meaning that the density has the function $f(p) = 1$. Then, after selecting a specific coin where the random variable $P$ has the value $p$, toss the coin 9 times and count the number of heads. Let $X$ be the number of heads. The *conditional* distribution of $X$ given a specific value of $P=p$, is binomial, $X\,\mid\, P=p \sim \text{Binomial}(9,p)$. But the *unconditional* distribution of $X$ is not binomial: it is instead has a distribution formed by averaging together many different binomial distributions, each with $n=9$, but with different values of $p$. We will use simulation to try to guess the distribution of $X$. This block of code generates $B = 1,000,000$ random realizations of $P$ and $X$. We can then calculate sample statistics of these simulated values as estimates of the values that could be found using analytical methods. The columns are `n`, 9 for each row, `p`, a head probability selected uniformly between 0 and 1, and `x`, a single generated value of $X$ selected at random from a binomial distribution with `n` and `p` for each given row. ***`{r} set.seed(2022) B = 1000000 prob7 = tibble( n = rep(9, B), p = runif(B), x=rbinom(B, n, p) ### 7A Use geom_density() to display an estimated graph of the density of $P$. We know that the exact density is uniform between 0 and 1. Does the estimated density approximate the exact density well? ***{r} ### 7B Calculate the mean and standard deviation of the values in the column `p`.