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Statistics Question on Multivariate Distributions

Let 𝑋1, 𝑋2, be a sequence of 𝑖.𝑖.𝑑. random variables each having π‘ˆ(0,1) distribution. Let π‘Œ be a random variable having the distribution function 𝐺. Suppose that
lim𝑛nβ†’βˆž(π‘‹βˆ’1+𝑋2+𝑋𝑛4≀π‘₯)=𝐺(π‘₯)lim𝑛_{nβ†’βˆž} \frac{( 𝑋-1+𝑋_2+𝑋_𝑛}{4} ≀ π‘₯)=𝐺(π‘₯), for all π‘₯ ∈ ℝ .
Then, π‘‰π‘Žπ‘Ÿ(π‘Œ) equals

A

112\frac{1}{12}

B

132\frac{1}{32}

C

148\frac{1}{48}

D

164\frac{1}{64}

Answer

164\frac{1}{64}

Explanation

Solution

The correct option is (D): 164\frac{1}{64}