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 …

Question

asked Jun 21, 2021 69.0k views

1 Answer

LeBaptiste · Answer 1 · 2021-06-27T20:33:11+0000

Answer:

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 ⇒

'' 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 ⇒

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 ⇒

1 / λ ⇒

λ =
(1)/(8)

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

The cumulative distribution function of
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
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