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yKpoI14uk [10]
3 years ago
7

The commute time for people in a city has an exponential distribution with an average of 0.5 hours. What is the probability that

a randomly selected person in this city will have a commute time between 0.4 and 1 hours? Answer: (round to 3 decimal places)
Mathematics
1 answer:
mamaluj [8]3 years ago
4 0

Answer:

0.314 = 31.4% probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

f(x) = \mu e^{-\mu x}

In which \mu = \frac{1}{m} is the decay parameter.

The probability that x is lower or equal to a is given by:

P(X \leq x) = \int\limits^a_0 {f(x)} \, dx

Which has the following solution:

P(X \leq x) = 1 - e^{-\mu x}

The probability of finding a value higher than x is:

P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}

In this question:

m = 0.5, \mu = \frac{1}{0.5} = 2

What is the probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours?

P(0.4 \leq X \leq 1) = P(X \leq 1) - P(X \leq 0.4)

In which

P(X \leq 1) = 1 - e^{-2} = 0.8647

P(X \leq 0.4) = 1 - e^{-2*0.4} = 0.5507

So

P(0.4 \leq X \leq 1) = P(X \leq 1) - P(X \leq 0.4) = 0.8647 - 0.5507 = 0.314

0.314 = 31.4% probability that a randomly selected person in this city will have a commute time between 0.4 and 1 hours

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