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 20 minutes, what is the probability that x is less than 23 minutes? (do not round until the final step

Answer 1

Explanation:

the exponential distribution function is

λe^(-λx) for x > 0

0 for x <= 0

λ is the distribution rate.

the mean value is the expected value, which is the integral across the whole function from 0 to infinity.

and that is :

1/λ

we know in our case that

mean value = 1/λ = 20 minutes.

1 = 20λ

λ = 1/20 = 0.05

we also know that

P (x>t) = e^(-λt)

so,

P (x<t) = 1 - P(x>t)

that includes also equality to the specified number.

t = 23.

P (x < 23) = 1 - e^(-0.05×23) = 1 - e^-1.15 =

= 1 - 0.316636769... =

= 0.683363231... ≈ 0.68

the probabilty that x is less than 23 minutes is about 0.68