Question: 10. Let X1, X2, . . . and Y1, Y2, . . . be independent random variables each having mean and non-zero variance

10. Let X1, X2, . . . and Y1, Y2, . . . be independent random variables each having mean μ and non-zero variance σ 2. Show that Un =

1

√2nσ 2

Xn i=1 Xi −

Xn i=1 Yi

satisfies, as n → ∞, P(Un ≤ x) →

Z x

−∞

1

√2π

e−1 2 u2 du for x ∈ R.

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