(Solved): help me solve thisDo not use chat gpt 2. We will now derive the probability that a given observation ...

2. We will now derive the probability that a given observation is part of a bootstrap sample. Suppose that we obtain a bootstrap sample from a set of $n$ observations. (a) What is the probability that the first bootstrap observation is not the $j$ th observation from the original sample? Justify your answer. (b) What is the probability that the second bootstrap observation is not the jth observation from the original sample? (c) Argue that the probability that the $j$ th observation is not in the bootstrap sample is $(1?1/n{)}^n$. 20 5. Resampling Methods (d) When $n=5$, what is the probability that the $j$ th observation is in the bootstrap sample? (e) When $n=100$, what is the probability that the jth observation is in the bootstrap sample? (f) When $n=10,000$, what is the probability that the $j$ th observation is in the bootstrap sample? (g) Create a plot that displays, for each integer value of $n$ from 1 to 100,000 , the probability that the $j$ th observation is in the bootstrap sample. Comment on what you observe. (h) We will now investigate numerically the probability that a bootstrap sample of size $n=100$ contains the $j$ th observation. Here $j=4$. We repeatedly create bootstrap samples, and each time we record whether or not the fourth observation is contained in the bootstrap sample.