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 = 0.383 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?

a) There is a 9% probability that a drought lasts exactly 3 intervals.

There is an 85.5% probability that a drought lasts at most 3 intervals.

b)There is a 14.5% probability that the length of a drought exceeds its mean value by at least one standard deviation

Step-by-step explanation:

The geometric distribution is the number of failures expected before you get a success in a series of Bernoulli trials.

It has the following probability density formula:

In which p is the probability of a success.

The mean of the geometric distribution is given by the following formula:

The standard deviation of the geometric distribution is given by the following formula:

In this problem, we have that:

So

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

This is

There is a 9% probability that a drought lasts exactly 3 intervals.

At most 3 intervals?

This is

Previously in this exercise, we found that

So

There is an 85.5% 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?

This is .

We are working with discrete data, so 3.66 is rounded up to 4.

Either a drought lasts at least four months, or it lasts at most thee. In a), we found that the probability that it lasts at most 3 months is 0.855. The sum of these probabilities is decimal 1. So:

There is a 14.5% probability that the length of a drought exceeds its mean value by at least one standard deviation

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answer:   x = 3 + √11     and     x = 3 - √11

step-by-step explanation:

x² - 6x + 9 = 11

(x - 3)² = 11

x - 3 = ±√11

    x = 3 ± √11

x = 3 + √11     and     x = 3 - √11