Let Y₁ and Y₂ be independent normal random variables, each with mean 0 and variance o². Define U₁ = Y₁ + Y₂ and U₂ = Y₁ - Y₂. Show that U₁ and U₂ are independent normal random variables, each with mean 0 and variance 20². [Hint: If (U₁, U₂) has a joint moment-generating function m (1₁, 12), then U₁ and U₂ are independent if and only if m(1₁, 12) = mʊ₁ (1₁)mʊ₂ (1₂).]

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