Answer:
The shape and rate parameters are [tex]\frac{1}{12}[/tex] and [tex]3[/tex].
Step-by-step explanation:
Let X = service time for each individual.
The average service time is, β = 12 minutes.
The random variable follows an Exponential distribution with parameter, [tex]\lambda=\frac{1}{\beta}=\frac{1}{12}[/tex].
The service time for the next 3 customers is,
Z = X₁ + X₂ + X₃
All the X[tex]_{i}[/tex]'s are independent Exponential random variable.
The sum of independent Exponential random variables is known as a Gamma or Erlang random variable.
The random variable Z follows a Gamma distribution with parameters (α, n).
The parameters are:
[tex]\alpha =\lambda=\frac{1}{12}\\n=3[/tex]
Thus, the shape and rate parameters are [tex]\frac{1}{12}[/tex] and [tex]3[/tex].
