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In the case of discrete random variables, it is quite clear that probability measure (or pmf) is quite useful since it gives the direct probability value.

But in case of a continuous random variable, in my opinion, the practicality of probability measure (valued after integrating pdf within certain limits) is not that much significant and pdf can be used instead of probability measure.

Is my opinion correct?

Suppose I have a continuous distribution of human heights, then if I want a measure for a chance of height = 7.889. Then obviously probability measure will be of no use. So, can I use the pdf only in this case as a measure.

In this aspect what is the importance of probability measure since we are not much interested in ranges?

hanugm
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    In general, the probability measure can be useful, of course. E.g. if you're interested in the probability of being at least 6ft.

    I think in your example there is some clash between discrete and continuous measures: The specific value 7.889 is one choice of uncountably many values and therefore (in general) has probability zero - so, technically, the question is not appropriate. However, I suppose implicitly you'd always round to at most 3 digits, so that the "next" value considered would be 7.89 and you end up with a discrete distribution based on intervals of length 0.001.

     – Mau314 Aug 23 '19 at 06:12
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    This is also what happens in real life, as we simply cannot measure with infinite precision. As humans, we have no choice but to be interested in ranges. – Mau314 Aug 23 '19 at 06:13

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