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 26 minutes, what is the probability that X is less than 30 minutes? (Do not round until the final step. Round your answer to 3 decimal places.)

Answer 1

Answer:

0.685 = 68.5% probability that X is less than 30 minutes

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:

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

If X has an average value of 26 minutes

This means that $m = 26, \mu = \frac{1}{26}$

What is the probability that X is less than 30 minutes?

$P(X \leq 30) = 1 - e^{-\frac{30}{26}} = 0.685$

0.685 = 68.5% probability that X is less than 30 minutes