Suppose that X_1, i= 1, 100, are random variables measuring the annual rainfall

Question

Suppose that X_1, i= 1, 100, are random variables measuring the annual rainfall (in inches) in a part of the Sahara dessert, over 100 consecutive years. Assume that these are independent and identically distributed, each having a mean of 1 inch and a standard deviation also of 1 inch. (a) First assume that the random variables X_i are Gaussian (ignoring the fact that rainfall cannot be negative). Find the probability that the total rainfall over the 100 year period exceeds 120 inches. (b) Suppose you do not know the underlying distributions of the random variables. Using Chebyshev's inequality, determine a bound on the probability that the total rainfall exceeds 120 inches.

Explanation / Answer

p = prob for any student declined = 0.3

q = prob for any student admitted = 0.97

and each student is independent.

Poisson mean = np = 900 (given)

Hence n =900/0.3 = 3000

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No of students admitted = 3000

Prob for an admitted student to graduate = 0.9

Hence mean of Poisson = np = 3000(0.9) = 270

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