In a Poisson process with a geometric random variable t, the arrival instant is not uniformly distributed in the interval [0,t]. The conditional probability of an arrival occurring before a specific time τ, given that an arrival has occurred before t, does not yield a uniform distribution. The calculations show that the probability of at least one arrival in [0,t] is influenced by the geometric nature of t. Thus, while individual events may appear uniformly distributed in deterministic intervals, the randomness of t alters this uniformity. Therefore, the conclusion is that the arrival instant is not uniform in the context of a geometric random variable.