The random variables X 1 , , X n are i i d We also know that E X 1 0, E X 1 2 a and E X 1 3 b and E X 1 4 c Let X n n X 1 X n Find the fourth moment of X n E x n 4 n 4 1 E i 1 n x i 4 i 1 n j i 4x i 3 x j i 1 n j i 6x i x j i 1 n j i k j , i 12x i 2 x j x k Please provide details of computation using above notation

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Question: The random variables X 1 ,..., X n are i.i.d. We also know that E[X 1 ] =0, E[X 1 2 ]=a and E[X 1

The random variables X₁,..., X_n are i.i.d. We also know that E[X₁] =0, E[X₁²]=a and E[X₁³]=b and E[X₁⁴]=c. Let X_n=nX1+...+Xn. Find the fourth moment of X_n .

E[x_n⁴]= n41E[i=1n x_i⁴ + i=1n j=i 4x_i³x_j + i=1n j>i 6x_ix_j +i=1n j=i k=j,i 12x_i²x_jx_k]

Please provide details of computation using above notation.

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