For a certain river, suppose the drought length Y is the number of consecutive t

Question

For a certain river, suppose the drought length Y is the number of consecutive time intervals in which the water supply remains below a critical value y0 (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). An article proposes a geometric distribution with p = 0.387 for this random variable. (Round your answers to three decimal places.)

(a) What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals? exactly 3 intervals at most 3 intervals

(b) What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?

Explanation / Answer

(a) P(exactly 3 intervals) = 0.387 * (1 - 0.387)^3 = 0.089

P(at most 3 intervals) = P(0 intervals) + P(1 interval) + P(2 intervals) + P(3 intervals)

[ 0.387* (1 - 0.387)^0] + [ 0.387 * (1 - 0.387)^1] + [ 0.387* (1 - 0.387)^2 ] + [ 0.387 * (1 - 0.387)^3]

= 0.86

(b) mean = (1-p)/p = 0.613/0.387 = 1.584

standard deviation = sqrt[(1-p)/p^2] = sqrt(0.613 / 0.387^2) = sqrt(4.093) = 2.023

P(Y > (1.584+ 2.023 )) = P(Y > 3.603) = P(Y at least 4) = 1 - P(Y at most 3)

= 1 - 0.86

= 0.14

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