Understanding the Probability in Continuous Distributions: Comparing P(X ≤ a) to P(X < a)

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

In continuous distributions, the probability that a random variable “x” is less than or equal to a certain value “a” is the same as the probability that “x” is strictly less than “a”

In continuous distributions, the probability that a random variable “x” is less than or equal to a certain value “a” is the same as the probability that “x” is strictly less than “a”. This is due to the nature of continuous distributions, which involve uncountably infinite outcomes.

To understand this concept better, let’s consider the cumulative distribution function (CDF) of a continuous random variable. The CDF gives the probability that “x” takes a value less than or equal to a given value. Formally, for a continuous random variable, the CDF can be expressed as P(X ≤ a).

Since “x” is a continuous random variable, we can think of the probability for any specific point as zero. In other words, the probability of “x” taking on a particular value “a” is extremely small, so it can be considered negligible.

Therefore, in continuous distributions, the probability can be represented as P(X < a) instead of P(X ≤ a). This is because the probability of "x" taking on any specific value (e.g., "a") is essentially zero. Thus, the probability that "x" is strictly less than "a" includes all the possible values that "x" can take, and hence is equivalent to the probability that "x" is less than or equal to "a". In summary, for continuous distributions, the probability that "x" is less than or equal to a certain value "a" is the same as the probability that "x" is strictly less than "a". This is because the probability of "x" taking a specific value in a continuous distribution is essentially zero.

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