---
title: "Short note: Convolution in random number"
subject: "Simulation and Modeling"
chapter: 6
key: "semester_8/simulation_and_modeling:convolution-short-note"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/convolution-short-note
site: BCT Engineering
---

# Short note: Convolution in random number

Simulation and Modeling · Chapter 6: Random Number

- Asked in: 2070 Bhadra, 3 marks
- Video: [Random Variate Generation (Part 3 Convolution Technique)](https://www.youtube.com/watch?v=t9b_wIWuJZ4) — Jayendran Venkateswaran
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/convolution-short-note
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-6

## Question

Write a short note on convolution in random number.

## Short answer

Convolution generates a variate as the sum of independent random variables. Erlang‑k: Y = −(1/kθ)ln(R1R2…Rk). Normal: Z = sum of 12 U(0,1) minus 6. Binomial: sum of Bernoulli trials. Advantages: simple when the inverse transform of the sum is hard. Disadvantage: needs many random numbers per variate.

## Answer

**Convolution** is a method of generating a random variate that is itself the **sum of two or more independent random variables**. If Y = X1 + X2 + ... + Xk, where the Xi are independent, then a value of Y is obtained by generating one value of each Xi (from random numbers) and adding them. The distribution of Y is the convolution of the distributions of the Xi.

**Examples**
- **Erlang-k distribution**: the sum of k independent exponential variables with mean 1/(kθ). Generate Xi = −(1/kθ) ln Ri, then Y = Σ Xi = −(1/kθ) ln(R1R2...Rk).
- **Binomial(n, p)**: the sum of n independent Bernoulli trials. For each trial, generate R; if R ≤ p count 1, else 0. Then add.
- **Normal variate**: by the central limit theorem, the sum of 12 uniform random numbers minus 6 is approximately N(0, 1): Z = R1 + R2 + ... + R12 − 6. Then X = μ + σZ.
- **Poisson / gamma** variates can similarly be built from sums of exponential variables.

**Advantages:** simple to understand and program, and applies when the inverse transform of the sum is hard to find. **Disadvantage:** needs k random numbers for each variate, so it is slow when k is large.

## Check yourself

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

1. Convolution generates a random variate that is:
   A. The product of dependent variables
   B. The sum of two or more independent random variables
   C. The maximum of uniform numbers
   D. The inverse of the cdf
   Answer: B. If Y = X1 + ... + Xk with independent Xi, one value of each is generated and added. (Key point: Convolution idea)

2. How is an Erlang-k variate generated by convolution?
   A. Generate one exponential and multiply by k
   B. Sum k independent exponential variates, Y = -(1/k theta) ln(R1 R2 ... Rk)
   C. Add k Bernoulli trials
   D. Add 12 uniforms and subtract 6
   Answer: B. Erlang-k is the sum of k independent exponentials, each -(1/k theta) ln Ri. (Key point: Erlang example)

3. How is an approximately N(0,1) variate obtained by convolution?
   A. Sum of 12 uniform random numbers minus 6
   B. Product of 12 uniforms
   C. Sum of 6 uniforms minus 12
   D. Square root of one uniform
   Answer: A. By the central limit theorem Z = R1 + ... + R12 - 6 is approximately N(0,1). (Key point: Normal approximation)

4. What is the main disadvantage of convolution?
   A. It is hard to program
   B. It needs k random numbers per variate, so it is slow for large k
   C. It only works for uniform variables
   D. It gives biased means
   Answer: B. Each variate needs k random numbers, which is slow when k is large. (Key point: Disadvantage)

## Related in Simulation and Modeling

- [Random number generation methods](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-generation-methods) — Simulation and Modeling, chapter 6
- [Random number; problems in generating pseudo-random](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-problems-pseudo-random) — Simulation and Modeling, chapter 6
- [Random number properties and Gap test](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-gap-test) — Simulation and Modeling, chapter 6, asked 4×
- [Random number properties; KS test algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-ks-algorithm) — Simulation and Modeling, chapter 6