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General Question, I wonder why every question in a probability exercise where we need to check the distribution of a random variable it always goes by showing the cdf and where it's defined. why we don't show pmf or pdf instead? of course, we can conclude it either way.. is it just a common habit?

Thanks.

Harel Yacovian
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    Because not all cdfs have a corresponding pdf. For example, take the cdf $P[X \le \alpha] = 1_{[0,\infty)}(\alpha)$. – copper.hat Oct 01 '18 at 20:28

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For the continuous distributions the probability is the area bounded by $ pdf$ and the x-axis and two given values.

$$ P(a\le x \le b) = \int _a ^b f(t) dt = cdf(b)-cdf(a) $$

Thus the $pdf$ by itself does not represent probabilities.

Mohammad Riazi-Kermani
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