Is this particular Student-t Distribution defined for a realization of standard deviation which is equal to zero? Given the student-t distribution:
$$
 T  =  \frac{\bar{X} - \mu}{S/\sqrt{N}}
$$
where $S$ is the Sample Standard Deviation, I wonder if this distribution is defined for $S=0.$ Is there something I am missing?
 A: Normally one has
\begin{align}
& X_1,\ldots,X_n \sim \text{i.i.d} \operatorname N(\mu,\sigma^2) \\[8pt]
& \overline X = (X_1+\cdots+X_n)/n \sim \operatorname N(\mu,\tfrac{\sigma^2} n) \\[8pt]
& S^2/\sigma^2 = \big( (X_1-\overline X)^2 + \cdots + (X_n-\overline X)^2\big) \sim \chi^2_{n-1} \\[8pt]
& \frac{\overline X - \mu}{\sigma/\sqrt n} \sim \operatorname N(0,1) \\[8pt]
& \frac{\overline X-\mu}{S/\sqrt n} \sim t_{n-1}
\end{align}
In this scenario, the probability that $S=0$ is $0$ unless $\sigma=0,$ so there is no need to consider that.
If you apply this to samples from a discrete distribution rather than to a normal distribution, then there is nonzero probability that $S^2=0.$

*

*Is the value of the random variable defined for $S=0$? Not unless one might have some reason to say it is $\infty$ or something like that.

*Is the distribution, defined in that case? Certainly even in the case of the normal distribution, one would not define a t-distribution for any particular value of $S$, unless one speaks of a conditional distribution given a particular value of $S.$ There is the question of the asymptotic distribution as $n\to\infty,$ and if the population distribution is continuous, I think that's been examined in published papers. I don't know about the discrete case though.

