Suppose customers in a certain queue are served one at a time sequentially, that the time for each individual customer is Exponential, and the average service time per customer equals 12 minutes. (Assume service times are independent of each other.) The time, in minutes, until the next 3 services are completed is Gamma (more specifically, Erlang) with what shape and rate parameters?

The shape and rate parameters are and .

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, .

The service time for the next 3 customers is,

Z = X₁ + X₂ + X₃

All the X'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:

Thus, the shape and rate parameters are and .

75 tickets have been sold so far.

step-by-step explanation:

in this question it is clearly mentioned that 45 children's tickets have been sold and this number is 60% of the total tickets sold.

now let's assume the total number of tickets sold be x.

then 60% of x = 45

⇒

⇒ x(.6) = 45

by dividing with .6 on both the sides of the equation

⇒ x = 75

therefore 75 tickets have been sold so far.

the answer is y < -3 or   y > 3

-3y > 7 + 2

-3 y > 9

y < -3