正态分布与伽马分布的卷积是什么？是否存在闭式表达式？

Great questions—let’s break this down clearly:

1. What is the convolution of a Normal Distribution (ND) and Gamma Distribution (GD)? #

First, let’s ground this in random variable terms: if you have two independent random variables ( X \sim \mathcal{N}(\mu, \sigma^2) ) (normal) and ( Y \sim \text{Gamma}(k, \theta) ) (shape parameter ( k ), rate parameter ( \theta )), their convolution is the probability density function (PDF) of the sum ( Z = X + Y ).

Mathematically, the convolution PDF ( f_Z(z) ) is defined as:
$$
f_Z(z) = \int_{-\infty}^{\infty} f_X(z - y) f_Y(y) dy
$$
Since the Gamma distribution only has support on non-negative real numbers, we can restrict the integral to ( y > 0 ). Substituting the respective PDFs gives:
$$
f_Z(z) = \int_{0}^{\infty} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(z - y - \mu)^2}{2\sigma2}\right) \cdot \frac{\theta^k}{\Gamma(k)} y^{k-1} \exp(-\theta y) dy
$$

2. Is there a closed-form expression for this convolution? #

Unfortunately, there’s no general closed-form solution using elementary functions for this convolution. The integral combines the normal distribution’s exponential-quadratic term with the gamma distribution’s power-law and exponential term, and this combination can’t be simplified to a basic set of standard functions (like polynomials, exponentials, or error functions) for arbitrary ( k ).

That said, there are a few special cases where you can get a more tractable form:

• If ( k ) is a positive integer, you can expand the integral into a series and express the result using the error function ( \text{erf}(\cdot) ) paired with polynomial terms.
• For small fractional values of ( k ) (e.g., ( k = 1/2 ), which maps to a scaled chi-square distribution), you might use specialized functions like the Dawson function, but this still isn’t a "simple" closed form.

In most practical scenarios, people rely on numerical integration or approximation techniques (like saddle-point approximations) to compute this convolution when needed.

内容的提问来源于stack exchange，提问作者Carl