The weather in a certain locale consists of alternating wet and dry spells. Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are independent. What is the long-run fraction of time that it rains?
Answers
Answer:
2/9
Step-by-step explanation:
The Poisson’s distribution is a discrete probability distribution. A discrete probability distribution means that the events occur with a constant mean rate and independently of each other. It is used to signify the chance (probability) of a given number of events occurring in a fixed interval of time or space.
In the long run, fraction of time that it rains = E(Number of days in rainy spell) / {E(Number of days in a rainy spell) + E(Number of days in a dry spell)}
E(Number of days in rainy spell) = 2
E(Number of days in a dry spell) = 7
In the long run, fraction of time that it rains = 2/(2 + 7) = 2/9
Final answer:
Given the parameters of the rainy spell and dry spell, the long-run fraction of time that it rains can be calculated by dividing the mean of the rainy days by the sum of the average rainy and dry days. Hence, it rains roughly 22.22% of the time in the long-term.
Explanation:
The question is asking about the long-run fraction of time that it rains, based on a rainy spell following a Poisson distribution with a mean of 2 days, and a dry spell following a geometric distribution with an average of 7 days, with the sequences being independent.
We are being asked to calculate the proportion of time that it rains in the long-run, given these distribution parameters. The Poisson and geometric distributions are often used in this type of probability assessment.
To tackle this, we need to divide the mean of the rainy days by the sum of the average rainy and dry days. Thus, the long-run fraction of time it rains is given by
So, in the long run, it rains roughly 22.22% (or 2/9) of the time.
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