Consider the COVID-19 infection and recovery cases from a start date of March 1, 2021. Define day i in the range [0, 365], inclusive, to be the number of days since this start date, and define the following: . Ci, a random variable for the number of people currently sick/infected with COVID-19 on Day i. Let pc,i be the probability of a person being sick/infected with COVID-19 Day i; a value; assume Pe, is known for all i. . Ri, a random variable for the number of newly recovered cases on Day i. Let pr,; be the probability of recovery for an infected person on Day ; a value; assume pr,; is known for all i. . n is the population of the world; a value. Assume that (1) everyone who has recovered from COVID-19 will not be reinfected and/or re-recover, and (2) the population of the world n is a fixed value for all days i. In practice neither of these assumptions hold, but they greatly simplify our computations below. (a) (2 points) Consider the random variable C; for a particular day i. Note that C; is not a Binomial random variable because trials are not independent; two people in close proximity to an infected individual are more likely to get infected together. However, C; can still be defined as a sum of n Bernoulli random variables as follows: For each of the n individuals in the world, count 1 if the individual is infected (with probability Pc,), and 0 otherwise. Compute the expectation E[C4]. (b) (2 points) Define "aggregate case load" as the total sum of cases for all days, including repeat values of people that are sick for multiple days. That is, say there are only two people in our simplified example: Person A and Person B. If Person A is sick for Day 1/2 and Person B is sick for Day 2/3/4/5, the "aggregate case load" from Day 1-5 is 6. Using the above definitions, write an expression to define a new variable C in terms of existing variables, such that C represents the aggregate case load from March 1, 2021 to March 1, 2022. (c) (2 points) Compute the expectation of C, the variable you defined in part (b), E[C] in terms of pc,; for all i in the range [0, 365] and the population of the world, n. (d) (2 points) Suppose we derive the variance of C as follows: 365 Var(C) = Var(Ci) 1=0 What is a potential issue with this approach