In a continuous distribution, the probability that the sample mean equals the population mean is:

In a continuous distribution, the probability that the sample mean equals the population mean is indeed 0. This stems from the nature of continuous distributions, where there are infinitely many possible values that a random variable can take. Because of this infinite possibility, the likelihood of any specific exact value—such as the sample mean being exactly equal to the population mean—is effectively zero.

To put it in perspective, you can think of it as trying to hit a specific point on a line that stretches infinitely: no matter how precise your aim, the chance of landing exactly on that point is infinitesimal. This concept is grounded in the principles of probability theory and helps to illustrate the behavior of sample means in statistical analysis.

In the case of discrete distributions, outcomes are countable and could have non-zero probabilities for specific values, but continuous distributions operate under different rules that maintain the probability at zero for exact matches. Thus, understanding the properties of continuous distributions is essential in grasping why the probability of equality for the sample mean and the population mean is zero.