Suppose the time a child spends waiting at for the bus as a school bus stop is exponentially distributed with mean 6 minutes. Determine the probability that the child must wait between 4 and 5 minutes on the bus on a given morning.

P [ 4 < x < 5 ] = 0,0788 or 7,88 %

Step-by-step explanation:

We are going to solve an exponentially distributted problem

as μ = 6 then λ = 1/6

We are looking for the probability of:

P [ 4 < x < 5 ]

That probability is equal to:

P (x < 5) - P (x < 4) (1)

P (x < 5) = 1 - е∧ (-1/6) (5) ⇒ 1 - е∧ ( - 5/6)

and

P (x < 4) = 1 - е∧ (-1/6) (4) ⇒ 1 - е∧ ( - 4/6)

By substitution in equation (1)

P [ 4 < x < 5 ] = 1 - е∧ ( - 5/6) - [1 - е∧ ( - 4/6) ]

P [ 4 < x < 5 ] = е∧ ( - 4/6) ] - е∧ ( - 5/6)

Solving with excel

P [ 4 < x < 5 ] = 0,5134 - 04346

P [ 4 < x < 5 ] = 0,0788 or 7,88 %