# Cumulative distribution function (same as 26793)

A cdf Fx is stochastically greater than a cdf Fy if (1) Fx(t)<= Fy(t) for all t, and (2) there exists some t for which Fx(t) < Fy(t)

(a)show that if Fx is the cdf of X and Fy is the cdf of Y, then (1) P(X>t) >= P(Y>t) for all t and (2) P(X>t) > P(Y>t) for some t.(in other words, X tend to be bigger than Y) Give an example.

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#### Solution Preview

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First, let us have some basic definitions:

The cumulative distribution function (cdf) is the probability that the variable takes a value less than or equal to x. That is

F(x) = Pr [ X <= x ] = alpha

For a continuous distribution, this can be expressed mathematically as

F(x) + Integral (from -inf. to x) [f(y)dy]

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For a continuous function, the probability density ...