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Given that PDF value $_()$ for a particular $=_1$ does not have any probabilistic meaning (by definition $(=_1)=0$). We still see the use of $_(_1)$ as its likelihood.

My questions are:

  • What is the intuition behind using $_()$ as the likelihood ? In general, what is the interpretation of $_()$.
  • Am I correct if I say that $_(_1)$ holds significance only for comparison purposes with $_(_)$,$≠1$ and not otherwise?
Aroonalok
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1 Answers1

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The intuitive way to interpret f(x) is through its integral. If you integrate f(x) over the interval x=a to x=b, then the result is the probability of x falling into [a,b].

So its not quite the probability that X=x, but its very closely related. Specifically, you can interpret f(x) as being a relative probability. So for example if f(1) = 0.8 and f(2) = 1.6 then this is saying that x is twice as likely to be "very close" to 2 as it is to be "very close" to 1. Here, saying that x is 'very close' to some number M should be interpreted in terms of the limiting probability of falling into the interval [M-delta,M+delta] as delta->0

James
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