---
title: "Gamma: power supply inadequate"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:gamma-power-supply-12-million"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-power-supply-12-million
site: BCT Engineering
---

# Gamma: power supply inadequate

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2071 Chaitra, 5 marks; 2074 Asoj, 5 marks (asked 2 times)
- Video: [Gamma Distribution - Electricity Consumption Example](https://www.youtube.com/watch?v=YcT5aDjZ3sc) — StatsResource
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-power-supply-12-million
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

The daily consumption of electric power in a certain city follow a gamma distribution with α = 2 and β = 3. If the power plant of this city has daily capacity of 12 million kilowatt hours, what is the probability that this power supply will be inadequate on any given day?

## Short answer

For Gamma(2,3), inadequate supply is P(X>12). Integrate (x/9)e^(-x/3) from 12 to infinity; result is 5e^-4, about 0.0916.

## Answer

Let X = daily power consumption (million kWh), X ~ Gamma with α = 2 and β = 3, with pdf

f(x) = x^(α−1) e^(−x/β) / (βᵅ Γ(α)) = x e^(−x/3) / (9 · 1) = (x/9) e^(−x/3), x > 0 (since Γ(2) = 1, β² = 9).

The supply is inadequate when consumption exceeds the capacity of 12:

P(X > 12) = ∫₁₂^∞ (x/9) e^(−x/3) dx

Put u = x/3, so x = 3u and dx = 3 du. The limits become u = 4 to ∞:

P = ∫₄^∞ (3u/9) e^(−u) · 3 du = ∫₄^∞ u e^(−u) du

Integrating by parts, ∫ u e^(−u) du = −(u + 1) e^(−u):

P = [−(u + 1) e^(−u)]₄^∞ = (4 + 1) e⁻⁴ = 5 × 0.018316 = **0.0916**

**Answer: the probability that the supply is inadequate is 0.0916 (about 9.2 %).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The supply is inadequate when:
   A. Consumption is below 12
   B. Consumption equals 12
   C. Consumption exceeds 12, P(X > 12)
   D. The mean is above 12
   Answer: C. Capacity is 12 so demand over 12 is a shortfall. (Key point: Inadequate supply)

2. With alpha = 2, beta = 3 the pdf is:
   A. (x/3) e^(-x/9)
   B. x e^(-x/2)/3
   C. (x/6) e^(-x/3)
   D. (x/9) e^(-x/3)
   Answer: D. x e^(-x/beta) / (beta^2 Gamma(2)) with beta^2 = 9. (Key point: Pdf)

3. Putting u = x/3 changes the limits to:
   A. 12 to infinity
   B. 4 to infinity
   C. 0 to 4
   D. 3 to infinity
   Answer: B. x = 12 gives u = 4. (Key point: Substitution)

4. P(X > 12) equals:
   A. 4 e^-4
   B. 5 e^-4 = 0.0916
   C. e^-4
   D. 1 - 5 e^-4
   Answer: B. Integral of u e^-u from 4 is (4+1) e^-4. (Key point: Result)

## Related in Probability and Statistics

- [Gamma: water supply inadequate](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Agamma-water-supply-9-million) — Probability and Statistics, chapter 3

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