Understanding Probability for Continuous Random Variables: Why the Probability of a Specific Value is Typically Zero

The probability that a continuous random variable equals a specific value is always equal to 1

That statement is not correct

That statement is not correct. The probability that a continuous random variable equals a specific value is typically zero.

In a continuous probability distribution, the probability of any single point is infinitesimally small, because there are an infinite number of possible values that the random variable can take on. Instead, the probability is defined over intervals.

For example, let’s consider a continuous random variable X that follows a normal distribution. The probability of X equaling a specific value, say x = 2, is essentially zero. However, we can calculate the probability of X falling within a certain range or interval, such as P(1 < X < 3), which would give a non-zero probability. In summary, for continuous random variables, the probability of the variable taking on a specific value is effectively zero, while the probability is defined over intervals.

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