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May the cumulative distribution function be used to calculate the inter-quartile range and vise-versa?

It should be possible. Inter-quartile range is given by the difference between third quartile and first quartile, i.e $IQR=Q_3-Q1$ Now according to cumulative distribution function of R.V X, $P(X\leq ...
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Can a sequence of PMFs converge to a PDF?

It is possible, but the big question is when you can interchange limits Yes, this kind of convergence can definitely happen. For example, consider the sequence of discrete uniform distributions $\{ ...
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