The time it takes a worker on an assembly line to complete a task is exponentially distributed with a mean of 8 minutes. What is the probability that it will take a worker less than 4 minutes to complete the task

The probability is

Step-by-step explanation:

We know that the time it takes a worker on an assembly line to complete a task is exponentially distributed with a mean of 8 minutes.

Let's define the random variable ⇒

'' The time it takes a worker on an assembly line to complete a task ''

We know that is exponentially distributed with a mean of 8 minutes ⇒

~ Exp (λ)

Where '' λ '' is the parameter of the distribution.

Now, the mean of an exponential distribution is ⇒

1 / λ  (I)

We have the value of the mean '' '' , then we replace that value in the equation (I) to obtain the parameter λ ⇒

1 / λ  ⇒

λ =

Then ,   ~

The cumulative distribution function of is :

when and

1 - e ^ ( - λx) when (II)

If we replace the value of the parameter in (II) :

 when

We need to calculate

Given that is a continuous random variable :

We use the cumulative distribution function to calculate the probability :

The probability is

about 0.75 hours

step-by-step explanation: