Understanding Probability in Continuous Distributions: The Equivalence of ≤ and <

For continuous distributions, the probability that x is less that or equal to a value is the same as the probability that x is less than the value

For continuous distributions, such as the normal distribution or the exponential distribution, the probability that a random variable (denoted as X) is less than or equal to a specific value (denoted as a) is the same as the probability that X is less than the value a

For continuous distributions, such as the normal distribution or the exponential distribution, the probability that a random variable (denoted as X) is less than or equal to a specific value (denoted as a) is the same as the probability that X is less than the value a.

This can be mathematically represented as:

P(X ≤ a) = P(X < a) In other words, the probability that X is less than or equal to a particular value is equal to the probability that X is strictly less than that value. This property arises because continuous distributions deal with an infinite number of possible values. Since the likelihood of obtaining a specific value in a continuous distribution is infinitesimally small, the difference between including the equal sign or not in the probability calculation is negligible. For example, let's consider the standard normal distribution. Suppose we want to find the probability that a random variable X, following a standard normal distribution, is less than or equal to 1. This can be written as: P(X ≤ 1) = P(X < 1) Using statistical tables or software, we can find that the probability of a standard normal variable being less than 1 is approximately 0.8413. Therefore, the probability of X being less than or equal to 1 is also approximately 0.8413. In summary, for continuous distributions, the distinction between less than or equal to and strictly less than is generally negligible and can be considered as equal.

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