Question

8. In a certain city, the daily consumption of electric power in millions of kilowatt hours can be treated as a random variable having a gamma distribution with α = 3 and θ = 2.

If the power plant of this city has a daily capacity of 12 million kilowatt-hours, what is the probability that this power supply will be inadequate on any given day?

Note that there is no need to do an integration to answer the question.

Answer 1

Let X denote the consumption of power supply(in million of kilowatt hours) on any given day

Thus, X ~ Gamma(3, 2)

Since is an integer, the Gamma distribution is an Erlang distribution and is the probability distribution of the waiting time until the kth arrival in a one dimensional Poisson process with intensity = 1/2

Thus, P(X > x) = P(Y < )

where Y ~ Pois()

The probability that the power supply will be inadequate on any given day = P(X > 12)

= P(Y < 3)

Y follows Poisson distribution with parameter 12/2 = 6

Thus, P(Y < 3) = P(Y = 0) + P(Y = 1) + P(Y = 2)

=

= 0.062