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 = 0.365 for this random variable. (Round your answers to three decimal places.)

a. What is the probability that a drought lasts 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?

Answer 1

a). P(X = x)

=
`$p(1-p)^x$` for x = 0, 1, 2, ....

P(x ≤ 3) = 0.837

b). Expectation =
`$((1-p))/(p)$`

= 1.7397

Variance =
`$((1-p))/(p^2)$`

= 4.7663726

Standard deviation = 2.1832

Therefore, mean + standard deviation

= 1.7397 + 2.1832

= 3.9229

`$P(x > 3.9229) = 0.1626$`

So the required P = 2 x 0.1626

= 0.325