For a continuous random variable, why does $P(a < Z < b) = P(a \leq Z < b) = P(a < Z \leq b) = P(a \leq Z \leq b)$

My textbook puts this in a sidebox with the heading “Note” and doesn’t explain why. Could you tell me why this statement holds?

$P(a < Z < b) = P(a \leq Z < b) = P(a < Z \leq b) = P(a \leq Z \leq b)$

Answer

Nothing formal to add to this, but an analogy that really helped me to understand this came from a calculus text. Imagine you have an iron pipe of a certain length and weight. And you wish to cut it into two pieces. If the pipe is say 1 m long you might want to cut it in half at the 0.5 mark. Now think of the pipe’s weight as some constant times the length of the pipe, (we assume that all cross-sections of equal length has the same weight).

Cutting the pipe in half at the 0.5 m mark – how much weight do you lose? Remember that the only cross-section you are removing is the 0.5 m mark itself. So what is the length of this cross-section? Consider that 0.49999999… is not apart of it, and neither is 0.5000000000…1, or any other point that is close to, but not equal to 0.5 – so the length of this cross-section is technically zero. Which means you’re not really removing any weight at all.

This would explain why $\leq$ and $<$ are basically the same for continuous variables – including or excluding the endpoint really doesn’t change anything – for any point you pick close to the endpoint, there is still an infinite amount of points between them.

Does this make any sense?

Attribution
Source : Link , Question Author : Person , Answer Author : Nick Cox

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