# Convolution of probability distributions

The convolution of probability distributions is the operation that gives the distribution of the sum of independent random variables: if X and Y are independent with distributions α and β, the distribution of X + Y is the convolution α ∗ β.<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup> In measure-theoretic terms, the convolution µ ∗ ν of two probability measures is defined as the distribution of X + Y for independent random variables X and Y having distributions µ and ν.<sup>[2](https://www.statslab.cam.ac.uk/~james/Lectures/pm.pdf)</sup> The operation is central to probability theory because it converts a question about sums of random quantities into a well-defined computation on their individual laws.

| Key fact | Detail |
|---|---|
| Defining property | The distribution of the sum S_n = X_1 + ... + X_n of independent random variables is the convolution µ_n = α_1 ∗ α_2 ∗ ... ∗ α_n of their distributions, computed inductively by µ_{j+1} = µ_j ∗ α_{j+1}.<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup> |
| Discrete formula | For independent integer-valued variables, m_3(j) = Σ_k m_1(k) · m_2(j − k).<sup>[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/07%253A_Sums_of_Random_Variables/7.01%253A_Sums_of_Discrete_Random_Variables)</sup> |
| Continuous formula | For independent variables with densities g and h, (g ∗ h)(z) = ∫ g(x) h(z − x) dx.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup> |
| Characteristic functions | The characteristic function of a sum of independent variables is the product of their characteristic functions: ψ_n(t) = φ_1(t) φ_2(t) ... φ_n(t).<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup> |
| Moments | For independent summands, E(S_n) is the sum of the expectations and Var(S_n) = Var(X_1) + ... + Var(X_n).<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup> |
| Algebraic structure | Convolution is commutative and associative, so f ∗ g ∗ h is unambiguous.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup> |
| Numerical practice | FFT-based convolution is faster for large numbers of quadrature points but more sensitive to rounding errors than direct convolution.<sup>[5](https://link.springer.com/article/10.1007/s11222-024-10514-5)</sup> |

## Formulas for sums

**Discrete case.** Suppose X and Y are independent integer-valued random variables with probability mass functions m_1 and m_2. To find the distribution of Z = X + Y, condition on the value of X: if X = k, then Z = z exactly when Y = z − k. Summing over the disjoint possibilities gives P(Z = z) = Σ_k P(X = k) · P(Y = z − k), which defines the convolution m_3 = m_1 ∗ m_2 with m_3(j) = Σ_k m_1(k) · m_2(j − k).<sup>[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/07%253A_Sums_of_Random_Variables/7.01%253A_Sums_of_Discrete_Random_Variables)</sup> The same partitioning argument works for dependent variables using the joint distribution, but it simplifies to a convolution only under independence.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup>

**Continuous case.** If X and Y are independent with densities g and h, the density of Z = X + Y is the convolution

\[ (g * h)(z) = \int_{D_z} g(x)\, h(z - x)\, dx, \]

where D_z is the set of values x for which both x and z − x lie in the support. When both variables are non-negative, D_z = [0, z] and the integral runs from 0 to z.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup> Determining the set D_z is often the most challenging step in applying the formula.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup>

**Characteristic functions.** The characteristic function φ_X(t) = E[e^{itX}] is the Fourier transform of a variable's law. For independent X and Y, φ_{X+Y}(t) = E[e^{it(X+Y)}] = E[e^{itX}] E[e^{itY}] = φ_X(t) φ_Y(t), where the factorization uses independence.<sup>[6](https://ocw.mit.edu/courses/12-864-inference-from-data-and-models-spring-2005/21b7909d277ff71c41fd0c557b97c243_statis_regr_2_3.pdf)</sup> By the convolution theorem, the [Fourier transform](https://www.edgepedia.org/fourier-transform) of a product of functions is the convolution of their Fourier transforms, so multiplication of characteristic functions corresponds exactly to convolution of distributions.<sup>[6](https://ocw.mit.edu/courses/12-864-inference-from-data-and-models-spring-2005/21b7909d277ff71c41fd0c557b97c243_statis_regr_2_3.pdf)</sup> This gives a direct way of calculating distributions of sums of independent random variables.<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup>

## Algebraic properties

Convolution of probability density functions, discrete or continuous, is commutative and associative: f ∗ g = g ∗ f and (f ∗ g) ∗ h = f ∗ (g ∗ h). These properties have short probabilistic proofs: commutativity follows from X + Y = Y + X and associativity from (X + Y) + Z = X + (Y + Z) for independent variables.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup> The distribution of the constant 0, the point mass δ, acts as an identity, f ∗ δ = f.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup>

For a sequence of independent and identically distributed variables with common density f, the sum Y_n = X_1 + ... + X_n has density f^{*n}, the n-fold convolution power of f, with the convention f^{*0} = δ.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup> When appropriately scaled and centered, the distribution of Y_n converges to the standard normal distribution as n grows; this is the central limit theorem.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)</sup>

## Moments of sums

Convolution acts simply on the first two moments. Expectation is additive for any variables, E(X + Y) = E(X) + E(Y), but variance is additive if the covariance is zero: Var(X + Y) = Var(X) + Var(Y).<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup><sup> • </sup><sup>[17](https://pages.stat.wisc.edu/~kdlevin/teaching/Fall2022/STAT340/lecs/L05_IndepCondBayes.html)</sup> For n independent summands, the cross terms E[X_i − E(X_i)][X_j − E(X_j)] vanish by independence, giving Var(S_n) = Var(X_1) + ... + Var(X_n).<sup>[1](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)</sup> In the Gaussian case the closure is complete: multiplying the characteristic functions of independent Gaussian variables produces a new Gaussian variable whose mean is the sum of the means and whose variance is the sum of the variances.<sup>[6](https://ocw.mit.edu/courses/12-864-inference-from-data-and-models-spring-2005/21b7909d277ff71c41fd0c557b97c243_statis_regr_2_3.pdf)</sup> This property underlies uncertainty propagation in metrology, where the sum of independent Gaussian inputs is treated as Gaussian.<sup>[7](https://nvlpubs.nist.gov/nistpubs/technicalnotes/nist.tn.1900.pdf)</sup>

## Worked examples

Two standard examples show the range of outcomes. Let X and Y be independent exponential variables with common rate λ. The convolution integral gives f_S(s) = ∫_0^s λe^{−λx} λe^{−λ(s−x)} dx = λ² s e^{−λs}, the gamma(2, λ) density.<sup>[8](https://data140.org/fa18/textbook/chapters/Chapter_19/01_Convolution_Formula)</sup> Now let U_1 and U_2 be independent uniform variables on (0, 1). The joint density surface is flat, so the density of the sum rises linearly from 0 to 1 and falls linearly from 1 to 2: a [triangular distribution](https://www.edgepedia.org/triangular-distribution) with height 1.<sup>[8](https://data140.org/fa18/textbook/chapters/Chapter_19/01_Convolution_Formula)</sup> Repeated convolution of a uniform distribution was historically important: Laplace's Théorie analytique considers drawing numbered balls with replacement and asks for the distribution of the sum after i drawings, a convolution of [discrete uniform distributions](https://www.edgepedia.org/discrete-uniform-distribution), a problem first posed and answered by Montmort and De Moivre.<sup>[9](https://www.maths.tcd.ie/EMIS/journals/JEHPS/decembre2012/Zabell.pdf)</sup>

## History

The convolution formula for the distribution of a sum appears in Laplace's work of the early 1780s. Anders Hald, a historian of statistics, lists "Laplace's Convolution Formula for the Distribution of a Sum, 1781" among the milestones of the field.<sup>[10](https://www.york.ac.uk/depts/maths/histstat/haldcontents.htm)</sup> According to the IEEE history of the convolution operation, Hald's account states that Laplace used the convolution operation in a memoir written in 1778 and published in the Mémoires de l'Académie Royale des Sciences de Paris in 1781, to find the distribution of a mean.<sup>[11](https://www.embs.org/pulse/articles/history-convolution-operation/)</sup> The chain of tools through which sums of variables were later analyzed ran from coefficient extraction (De Moivre) to generating functions (Euler, Laplace) to characteristic functions (Fourier, Lévy).<sup>[12](https://pure.au.dk/portal/en/publications/from-coefficients-to-distributions-demoivre-and-the-operational-v/)</sup>

## Numerical computation

Exact convolution is often impractical to carry out by hand, so numerical methods matter. The discrete Fourier transform and its inverse provide fast computation of convolutions of a finite number of independent univariate variables, and published work derives mathematical error bounds for the resulting cumulative distribution functions.<sup>[13](https://ideas.repec.org/a/spr/metcap/v22y2020i3d10.1007_s11009-019-09739-z.html)</sup>

**Accuracy versus speed.** A 2024 study comparing direct convolution with FFT-based convolution for left-tail probabilities of sums found that FFT-based convolution is much more sensitive to rounding errors, with an approximation error proportional to the machine epsilon of the floating-point precision used; the machine epsilon is approximately 1.19 × 10^{−7} for 32-bit floats, 2.22 × 10^{−16} for 64-bit floats, 1.93 × 10^{−34} for 128-bit floats, and 1.81 × 10^{−71} for 256-bit floats.<sup>[5](https://link.springer.com/article/10.1007/s11222-024-10514-5)</sup> FFT-based convolution approximates a rare-event probability well only when the probability is much larger than the machine epsilon.<sup>[5](https://link.springer.com/article/10.1007/s11222-024-10514-5)</sup> The tradeoff is computational: FFT-based convolution has a lower asymptotic cost and eventually outperforms direct convolution in runtime for sufficiently large numbers of quadrature points N.<sup>[5](https://link.springer.com/article/10.1007/s11222-024-10514-5)</sup> Refinements such as the aFFT-C method keep the FFT's speed while using a rigorous analysis of roundoff error propagation to detect and circumvent the error accumulation inherent to the Fourier transform.<sup>[14](https://www.sciencedirect.com/science/article/abs/pii/S0167947316300536)</sup>

## Applications and extensions

**Actuarial science.** Aggregate loss models treat total claims S_N = X_1 + ... + X_N, where the claim count N is itself random. The compound distribution of S_N is derived from convolutions of the severity distribution: F_{S_N}(s) = p_0 + Σ_{n=1}^{∞} p_n F_X^{*n}(s), where p_n = P(N = n) and F_X^{*n} is the n-fold convolution of the claim-size distribution.<sup>[15](https://openacttexts.github.io/LDAVer2/ChapAggLossModels.html)</sup> When N is Poisson and the claim sizes are gamma, the result is the [Tweedie distribution](https://www.edgepedia.org/tweedie-distribution), which has a point mass at zero and a continuous positive component and is widely used for insurance claims modeling.<sup>[15](https://openacttexts.github.io/LDAVer2/ChapAggLossModels.html)</sup>

**Free probability.** A parallel theory exists for noncommuting random variables. For selfadjoint variables x and y that are freely independent with distributions µ and ν, the distribution of x + y is described by an operation called free convolution, defined in analogy with the classical convolution of probability measures that corresponds to sums of classically independent variables.<sup>[16](https://www.cambridge.org/core/books/abs/lectures-on-the-combinatorics-of-free-probability/sums-of-free-random-variables/7D53E21A44EB43CAB93F8A4D10B36FF1)</sup>

## References

1. [S. R. S. Varadhan, Probability course notes, Chapter 3: Independent Sums, Courant Institute, NYU](https://math.nyu.edu/~varadhan/course/PROB.ch3.pdf)
2. [Probability and Measure lecture notes, University of Cambridge StatsLab](https://www.statslab.cam.ac.uk/~james/Lectures/pm.pdf)
3. [7.1: Sums of Discrete Random Variables, Grinstead and Snell, Statistics LibreTexts](https://stats.libretexts.org/Bookshelves/Probability_Theory/Introductory_Probability_%28Grinstead_and_Snell%29/07%253A_Sums_of_Random_Variables/7.01%253A_Sums_of_Discrete_Random_Variables)
4. [3.7: Transformations of Random Variables, Siegrist, Statistics LibreTexts](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables)
5. [A fast and accurate numerical method for the left tail of sums of independent random variables, Statistics and Computing (2024)](https://link.springer.com/article/10.1007/s11222-024-10514-5)
6. [Inference from Data and Models, MIT OpenCourseWare, 12.864, Spring 2005](https://ocw.mit.edu/courses/12-864-inference-from-data-and-models-spring-2005/21b7909d277ff71c41fd0c557b97c243_statis_regr_2_3.pdf)
7. [NIST Technical Note 1900](https://nvlpubs.nist.gov/nistpubs/technicalnotes/nist.tn.1900.pdf)
8. [The Convolution Formula, Prob 140 Textbook, UC Berkeley](https://data140.org/fa18/textbook/chapters/Chapter_19/01_Convolution_Formula)
9. [S. L. Zabell, De Morgan and Laplace, Electronic Journal for History of Probability and Statistics, Vol. 8 (2012)](https://www.maths.tcd.ie/EMIS/journals/JEHPS/decembre2012/Zabell.pdf)
10. [Contents, Anders Hald, A History of Mathematical Statistics from 1750 to 1930](https://www.york.ac.uk/depts/maths/histstat/haldcontents.htm)
11. [A History of the Convolution Operation, IEEE Pulse](https://www.embs.org/pulse/articles/history-convolution-operation/)
12. [From Coefficients to Distributions: De Moivre and the Operational View of Probability, Aarhus University](https://pure.au.dk/portal/en/publications/from-coefficients-to-distributions-demoivre-and-the-operational-v/)
13. [Error Bounds for Cumulative Distribution Functions of Convolutions via the Discrete Fourier Transform, Methodology and Computing in Applied Probability (2020)](https://ideas.repec.org/a/spr/metcap/v22y2020i3d10.1007_s11009-019-09739-z.html)
14. [Accurate pairwise convolutions of non-negative vectors via FFT, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0167947316300536)
15. [Chapter 7: Aggregate Loss Models, Loss Data Analytics, Second Edition](https://openacttexts.github.io/LDAVer2/ChapAggLossModels.html)
16. [Sums of free random variables, Lectures on the Combinatorics of Free Probability, Cambridge University Press](https://www.cambridge.org/core/books/abs/lectures-on-the-combinatorics-of-free-probability/sums-of-free-random-variables/7D53E21A44EB43CAB93F8A4D10B36FF1)
17. [STAT340 Lecture 05: Independence, Conditional Probability and Bayes’ Rule](https://pages.stat.wisc.edu/~kdlevin/teaching/Fall2022/STAT340/lecs/L05_IndepCondBayes.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Probability distributions › Compound, infinitely divisible, and convolved distributions › Convolutions of distributions*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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