6a) Each magazine that Charlie reads takes an amount of time that follows an exponential distribution with mean 10 minutes. The number of magazines that Charlie reads on any given day has a Poisson distribution with mean 2. Assume that each reading session always falls within a single day (he does not read past midnight).

Question

Furthermore, suppose that the number of magazines that Charlie reads on different days are independent, and that the lengths of time he takes to read each magazine is also independent of each other.

For simplicity, assume that he only reads 1 magazine at a time and that there are 30 days in each given month.

Let 𝑇 be the total number of minutes Charlie spends reading magazines in one month.

Hint: You may find the following useful. For i.i.d. random variables 𝑋𝑖 and a nonnegative random variable 𝐾 that is independent of all the 𝑋𝑖's:

𝐄[∑𝑖=1𝐾𝑋𝑖] = 𝐄[𝐾]𝐄[𝑋1]
𝖵𝖺𝗋[∑𝑖=1𝐾𝑋𝑖] = 𝐄[𝐾]𝖵𝖺𝗋[𝑋1]+(𝐄[𝑋1])2𝖵𝖺𝗋(𝐾).
Note: Double check your answer for the following two answer boxes. The problem below depends on these.

Find 𝐄(𝑇).
Find 𝖵𝖺𝗋(𝑇) .

6b. Use the central limit theorem to approximate the probability that the total number of magazines that Charlie reads in one full year (12 months of 30 days each) is between 3500 and 3600.

(Note that you are asked about the number of magazines, NOT the number of minutes of total reading time.)

(Please answer the problem as written.)

(Give an answer accurate to at least 3 decimal places.)

Answer 1

Please answer the 2nd part also:

Use the central limit theorem to approximate the probability that the total number of magazines that Charlie reads in one full year (12 months of 30 days each) is between 3500 and 3600 .

(Note that you are asked about the number of magazines, NOT the number of minutes of total reading time.)

(Please answer the problem as written.)

(Give an answer accurate to at least 3 decimal places.)

Answer 2

leter b is weird..... does anyone else get any other answer rather than approximately cero?