Distribution of maximum of normally distributed random variables

Question

I'm trying to find the closed-form CDF and PDF of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$.

My thought process so far: $$ \begin{align*} F_Y(y) &= \mathbb{P}(\max(X_1, ..., X_n) \leq y) \\ &= \prod_{i=1}^n F_{X_i}(y) \\ \implies f_Y(y) &= \frac{\partial}{\partial y} \prod_{i=1}^n F_{X_i}(y) \end{align*} $$ I'm not sure how to proceed from here, or whether this is even the correct approach. Any help is much appreciated.

Edit: The linked question pertains to IID random variables, whereas I am more interested in INID random variables.

Answer 1

Problems like this, where you want to differentiate the product of a bunch of functions that depend on your variable of interest, can be dealt with by logarithmic differentiation. Let $\Phi$ and $\phi$ denote the CDF and PDF of the standard normal distribution (respectively). Since the normal random variables in your question have the same variance you get:

$$\prod_{i=1}^n F_i(y) = \prod_{i=1}^n \Phi \Big( \frac{y-\mu_i}{\sigma} \Big) = \exp \Bigg( \sum_{i=1}^n \ln \Phi \Big( \frac{y-\mu_i}{\sigma} \Big) \Bigg).$$

Differentiating with respect to $y$ and applying the chain rule gives:

$$\begin{equation} \begin{aligned} f_Y(y) = \frac{d F_Y}{dy}(y) &= \Bigg( \frac{1}{\sigma} \sum_{i=1}^n \frac{\phi ( (y-\mu_i)/\sigma ) }{\Phi ( (y-\mu_i)/\sigma )} \Bigg) \exp \Bigg( \sum_{i=1}^n \ln \Phi \Big( \frac{y-\mu_i}{\sigma} \Big) \Bigg) \\[6pt] &= \Bigg( \frac{1}{\sigma} \sum_{i=1}^n \frac{\phi ( (y-\mu_i)/\sigma ) }{\Phi ( (y-\mu_i)/\sigma )} \Bigg) \Bigg( \prod_{i=1}^n \Phi \Big( \frac{y-\mu_i}{\sigma} \Big) \Bigg). \\[6pt] \end{aligned} \end{equation}$$

In the special case where $\mu = \mu_1 = \cdots = \mu_n$ this reduces to the well-known formula:

$$f_Y(y) = \frac{n}{\sigma} \cdot \phi \Big( \frac{y-\mu}{\sigma} \Big) \cdot \Phi \Big( \frac{y-\mu}{\sigma} \Big)^{n-1}.$$

Answer 2

We can express each $F_{X_i}$ in terms of an error function, and when differentiating via the product rule to get $f_Y$ we can re-obtain from each of those error functions in turn a Gaussian function. It depends what you mean by closed form -- if you leave the $\mu_i$ as symbolic parameters then you will have explicit sums and products over $i$. It will be in suitable form to compute numerically for given parameters if you have access to elementary and error functions.