Question

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

Answer 1

The probability is
`0.3935`

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 ⇒

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

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

`X` ~ Exp (λ)

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

Now, the mean of an exponential distribution is ⇒

`E(X)=` 1 / λ (I)

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

`8=` 1 / λ ⇒

λ =
`(1)/(8)`

Then ,
`X` ~
`Exp((1)/(8))`

The cumulative distribution function of
`X` is :

`F_(X)(x)=P(X\leq x)=0` when
`x<0` and

`F_(X)(x)=P(X\leq x)=` 1 - e ^ ( - λx) when
`x\geq 0` (II)

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

`P(X\leq x)=1-e^{-(x)/(8)}` when
`x\geq 0`

We need to calculate
`P(X<4)`

Given that
`X` is a continuous random variable :

`P(X<4)=P(X\leq 4)`

We use the cumulative distribution function to calculate the probability :

`P(X\leq 4)=F_(X)(4)=1-e^{-(4)/(8)}=0.3935`

The probability is
`0.3935`