Question

The random variable X is exponentially distributed, where X represents the time it takes for a person to choose a birthday gift. If X has an average value of 30 minutes, what is the probability that X is less than 38 minutes? (Do not round until the final step. Round your answer to 3 decimal places.)

Answer 1

0.718 = 71.8% probability that X is less than 38 minutes

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=(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_0f ({x)} \, dx`

Which has the following solution:

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

If X has an average value of 30 minutes

This means that
`m=30,\mu=(1)/(30)`

What is the probability that X is less than 38 minutes?

`P(X\leq 38)=1-e^{-(38)/(30) }`

0.718 = 71.8% probability that X is less than 38 minutes