The Probability of Exact Values for Continuous Random Variables | Explained

For any continuous random variable, the probability that the random variable takes on exactly a specific value is

zero

zero.

For continuous random variables, the probability that the random variable takes on any particular exact value is zero. This is because continuous random variables can take on an infinite number of values within a given interval, and each individual value has zero probability of occurring.

To illustrate this, let’s consider an example. Suppose we have a continuous random variable X that represents the height of adults in a population. The possible values for X can be any real number between a certain lowest and highest height. For simplicity, let’s say X can range from 150 cm to 200 cm.

Now, if we want to calculate the probability that a randomly selected adult has a height of exactly 170 cm, we need to consider the probability distribution of X. The probability distribution describes the likelihood of X taking on certain values.

In the case of continuous random variables, the probability distribution is represented by a probability density function (PDF), such as the normal distribution. The PDF assigns probabilities to intervals, rather than specific values.

So, in our example, we can calculate the probability that a randomly selected adult has a height between 170 cm and 171 cm using the PDF. However, the probability of exactly 170 cm would be zero. This means that the chance of randomly selecting an adult with an exact height of 170 cm is infinitesimally small.

In summary, for continuous random variables, the probability of an exact value is considered zero due to the infinite number of possible values within the interval. Instead, we focus on the probabilities associated with intervals.

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