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So I know that a cdf F is defined via F(a) = Pr[X ≤ a]. How do I show that the cdf F of a random variable X contains exactly the same information as the function defined via G(a) = Pr[X = a], by expressing F in terms of G and expressing G in terms of F.

-- Compute and plot the cdf for (i) X ∼ Geom(p), (ii) X ∼ Exp(λ ).

-- Identify two key properties that a cdf of any r.v. has to satisfy.

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    In general, F cannot be expressed in terms of G. In your example (ii), G=0 identically and there is no way to guess F from that information.2011-11-17
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    I changed the subject line. There is no such thing as a cumulative density function. The word "cumulative" contradicts the word "density".2011-11-17

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