MATH SHU 235 Columbia University Probability and Statistics Questions There are four questions in total, and the questions are in the attached file. Thank

MATH SHU 235 Columbia University Probability and Statistics Questions There are four questions in total, and the questions are in the attached file. Thank you. BI-WEEKLY ASSIGNMENT 4
(1) Consider a random variable X with E(X) = 3 and E(X 2 ) = 13. Use
Chebyshev’s inequality to show that
21
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25
(2) We consider a sequence of throws of an unbiased die. For a positive integer j
let Xj = 1 if the outcome of the j-th throw is 5 or 6, and Xj = 0 otherwise.
(a) Compute the mean and the variance of X1 .
(b) Let T denote the number of throws that were 5 or 6 in the first 1800
throws. Approximate the probability P (580 < T < 640). [It is given that ?(1) = 0.8413, ?(1.5) = 0.9332, ?(2) = 0.9773.] (3) Two groups of undergraduate students A and B each having 200 members are preparing to take an exam. Their scores are independent from each other’s. We know that the scores of the students that belong to group A are identically distributed with mean 9 and variance 1/6, whereas the scores of the students that belong to group B are identically distributed with mean 8.5 and variance 1/3. Let MA and MB be the averages of the two groups. Compute the probability that MA ? MB ? [0.5, 0.65]. It is given that ?(1) = 0.8413, ?(2) = 0.9773, ?(3) = 0.9987. [Attention! The exam hasn’t taken place yet. The scores of the students as well as MA , MB are random variables. After the exam, they will take exact values, and thus there will be no randomness. The question above refers to for MA ? MB refers to the state prior the exam.] (4) We participate in a game in which in every round we win 1$ with probability 3/5 and we lose 1$ with probability 2/5. The outcome of every round is independent of the past. Specify the values of n ? 30 such that the probability that after n rounds we are winning less than 2$ is smaller than 0.1. [It is given that ?(?1.282) = 0.1.] P (?2 ? X ? 8) ? 1 Purchase answer to see full attachment