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I got a question in statistics and probability, and i would like to have some kind of help in solving it.

Question

As you may have noticed, Dr. Mike says “right” in class, A LOT, and now Dr. Mike has made this into a CFC fund raiser. Here is how the game works: Each student is provided a buzzer device, and must press said buzzer each time Dr. Mike says “right” in class. Now, each student must press the buzzer within a set amount of time to get credit (and randomly pressing the button when Dr. Mike hasn’t said “right” does not count either), and assume there is some automated process deciding if each buzzer press was correct or not. If all students in the class correctly press the button, then Dr. Mike will donate a random amount of money to the CFC. If even one student does not press the button within the window, then that no donation is made. Your goal through the next several parts is to characterize the distribution of the total donation to the CFC.

Recall that Section (1) has 31 students, and Section (2) has 9 students. Assume that both sections operate independently of each other, and that Dr. Mike says “right” according to a Poisson process with a mean rate of $64$ “rights” per class period. Finally, assume that all students press the buzzer independently from each other, and independently over time and that each student presses the buzzer correctly $95\%$ of the time. Finally, for each correct buzzer push, Dr. Mike will donate $Z \sim \mathrm{exponential}(\beta)$ dollars to the CFC. The goal of this problem is to characterize the distribution of the total donation from both classes, called $D$.

  1. First, let $p_1$ and $p_2$ be the probability that every student in the respective section correctly presses the button. Find $p_1$ and $p_2$.

  2. Let $X_1$ be the number of “rights” in Section (1) and $X_2$ be the number of “rights” in Section (2). Similarly, let $Y_1$ and $Y_2$ be the number of correct buzzer presses in each respective section. Identify the distributions of $X_1, X_2$, propose suitable distributions for $Y_1\mid X_1$, and $Y_2\mid X_2$, and derive the distributions of $Y_1$ and $Y_2$, separately. Are $Y_1$ and $Y_2$ independent of each other?

  3. Derive the distribution of $Y_{\text{total}} = Y_1 + Y_2$. Note: If you didn’t get an answer for problem 1, assume $Y_1 \sim \mathrm{Poisson}(\lambda = 31)$ and $Y_2 \sim \mathrm{Poisson}(\lambda = 9)$. Incidentally, this is not the right answer to 1.

  4. Now that we have the distribution of $Y_{\text{total}}$, we need to connect this with the amount of money generated by the fund raiser. Suppose you are given $Y_{\text{total}}$. Derive the distribution of $D\mid Y_{\text{total}}$.

  5. Write down the joint distribution of $(D, Y_{\text{total}})$ and find $E[D]$. DO NOT attempt to find the distribution of $D$. If you didn’t get $2$, use $Y_{\text{total}} \sim \mathrm{Poisson}(\lambda = 123.456)$, which is not the correct answer to 2.

  6. Mrs. Mike (wife of Dr. Mike) thinks this is a terrible idea, and promises that Dr. Mike will sleep in the dog house for a week if the total donation ends up being more than $\$300$. Choose the average donation $\beta$ to maximize the donation, subject to the constraint that the probability that Dr. Mike sleeps in the doghouse is less than $0.05$.

Thank you.

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Welcome to Math.SE! A few guidelines on getting the best help from the community possible: What have you done so far? What do you know? If you can support your questions with answers to these two questions, you will be more likely to receive the best help. –  Arkamis Nov 28 '12 at 16:42
    
for part (1) , I'm thinking to use Binomial Distribution with n=64 and p=0.95 and then sum from 0 to 31(for p1) , and sum from 0 to 9(for p2) does it look good ? I'm confused , which distribution should i use ? –  james Nov 28 '12 at 17:14

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