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Let (X, Y ) have joint cdf F, and let G be the cdf of the...

Let (X, Y ) have joint cdf F, and let G be the cdf of the random variable X + Y . Show that F(x, x) ≤ G(2x) for all x ∈ R

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Answer #1

Let A = (X x, Y x)

Hence, P(A) = P(X x, Y x) = F(x, x)

Let, B = (X + Y 2x)

Hence, P(B) = P(X + Y 2x) = G(2x)

Note that, X x and Y x X + Y x + x = 2x

i.e. A B i.e. A B i.e. P(A) P(B) [By the property of probability] i.e. F(x, x) G(2x). [Proved].

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