Chi-square from paired Gaussian differences

Let $X_1,X_2,…,X_{20}$ be a random sample from the $N(5,2)$ distribution and let $Y_i=X_{2i}-X_{2i-1}$, $i=1,2,…,10$. Then $W=\dfrac14\sum_{i=1}^{10}Y_i^2$ has the