If arrivals follow a poisson distribution with mean 1.2 arrivals per minute, find the 75th percentile of waiting times until the next arrival (i. e., 75 percent below).

To solve this given problem, we make use of the formula:

e ^ ( - μ x) = (1 - nth percentile)

where the variables are:

μ is the mean value or population mean = 1.2 arrivals per minute

x is the value at the nth percentile = unknown value

nth percentile = 75th percentile or 75% = 0.75

Substituting all the given values into the equation to find for x:

e ^ ( - 1.2 x) = 1 - 0.75

e ^ ( - 1.2 x) = 0.25

Taking the ln of both sides:

- 1.2 x = ln 0.25

x = - (ln 0.25) / 1.2

x = 1.155

Therefore the 75th percentile is 1.155 arrivals per minute