Check the solution to "Based on historical data, on average, Florida experiences..."

parthe0o

Answered question

2021-12-02

Based on historical data, on average, Florida experiences one flooding event every 2.4 months. Assume that the time between flooding events can be modeled using an exponential distribution. What is the median number of months from now until Florida experiences its next flooding event?

Answer & Explanation

Rosemary McBride

Beginner2021-12-03Added 10 answers

Step 1
Solution:
Consider a random variable X which follow exponential distribution and it represents the time between flooding events.

X \sim E x p (\frac{1}{2.4})

The probability distribution of X is,

f_{x} (x) = \frac{1}{2.4} \exp (- \frac{x}{2.4}), x \geq 0

Step 2
Let the median number of months be a. Then we have,

P (X \leq a) = 0.5

\int_{0}^{a} f_{x} (x) d x = 0.5

\int_{0}^{a} \frac{1}{2.4} \exp (- \frac{x}{2.4}) d x = 0.5

{[\frac{\frac{1}{2.4} \exp (- \frac{x}{2.4})}{- \frac{1}{2.4}}]}_{0}^{a} = 0.5

{[- \exp (- \frac{x}{2.4})]}_{0}^{a} = 0.5

- [\exp (- \frac{a}{2.4}) - \exp (- \frac{0}{2.4})] = 0.5

[1 - \exp (- \frac{a}{2.4})] = 0.5

[\exp (- \frac{a}{2.4})] = 0.5

(- \frac{a}{2.4}) = \ln (0.5)

a = - 2.4 \ln (0.5)

a = 1.6636

Thus, the median number of months from now until Florida experiences its next flooding event is 1.67 months.

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