Independent solution

How to solve this Raw Moments of Rounded Variables question

Setup

Setup

Compute the continuous fourth moment directly from the uniform density. Nearest-integer rounding gives the two endpoint cells half the probability of each interior cell.

E[X4]=110∫010x4 dx=2000E[X^4]=\frac1{10}\int_0^{10}x^4\,dx=2000

Model

Model

Use probability one twentieth at the endpoints and one tenth at each interior integer to form the fourth raw moment of the rounded variable.

Pr⁡(Y=0)=Pr⁡(Y=10)=120,Pr⁡(Y=k)=110 (1≤k≤9)\Pr(Y=0)=\Pr(Y=10)=\frac1{20},\quad \Pr(Y=k)=\frac1{10}\ (1\le k\le9)

Compute

Compute

The continuous and rounded fourth moments are 2,000 and 2,033.3. Their absolute difference is 33.3.

E[Y4]=110∑k=19k4+10420=2033.3,∣E[Y4]−E[X4]∣=33.3E[Y^4]=\frac1{10}\sum_{k=1}^9k^4+\frac{10^4}{20}=2033.3,\quad |E[Y^4]-E[X^4]|=33.3

Answer

Answer

The absolute fourth-moment difference rounds to 33.

33(B)\boxed{33\quad\text{(B)}}