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Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

1. F(x) ≥ G(x) = 0
2. F(x) ≤ G(x) = 0
3. (F(x) ≥ G(x)) ∀ x ≥ 0
4. (F(x) ≤ G(x)) ∀ x ≥ 0

Correct answer: (F(x) ≥ G(x)) ∀ x ≥ 0

Solution

For x>=0, G(x)=P(2V<=x)=P(V<=x/2)<=P(U<=x)=F(x) since x/2<=x and U,V are identically distributed. Thus F(x)>=G(x) for all x>=0, so the stored F<=G is wrong.

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