Answer:
The probability is [tex]0.3935[/tex]
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 ⇒
[tex]X:[/tex] '' The time it takes a worker on an assembly line to complete a task ''
We know that [tex]X[/tex] is exponentially distributed with a mean of 8 minutes ⇒
[tex]X[/tex] ~ Exp (λ)
Where '' λ '' is the parameter of the distribution.
Now, the mean of an exponential distribution is ⇒
[tex]E(X)=[/tex] 1 / λ (I)
We have the value of the mean '' [tex]E(X)[/tex] '' , then we replace that value in the equation (I) to obtain the parameter λ ⇒
[tex]8=[/tex] 1 / λ ⇒
λ = [tex]\frac{1}{8}[/tex]
Then , [tex]X[/tex] ~ [tex]Exp(\frac{1}{8})[/tex]
The cumulative distribution function of [tex]X[/tex] is :
[tex]F_{X}(x)=P(X\leq x)=0[/tex] when [tex]x<0[/tex] and
[tex]F_{X}(x)=P(X\leq x)=[/tex] 1 - e ^ ( - λx) when [tex]x\geq 0[/tex] (II)
If we replace the value of the parameter in (II) :
[tex]P(X\leq x)=1-e^{-\frac{x}{8}}[/tex] when [tex]x\geq 0[/tex]
We need to calculate [tex]P(X<4)[/tex]
Given that [tex]X[/tex] is a continuous random variable :
[tex]P(X<4)=P(X\leq 4)[/tex]
We use the cumulative distribution function to calculate the probability :
[tex]P(X\leq 4)=F_{X}(4)=1-e^{-\frac{4}{8}}=0.3935[/tex]
The probability is [tex]0.3935[/tex]
