Expectation of ratio of normal and root chi-square

Question

Let $X_1,X_2,X_3, X_4$ be i.i.d $N(0,1)$ random variables. What is the expectation of $$(X_1-X_2+X_3)/\sqrt {X_1^2+X_2^2+X_3^2+X_4^2}$$? I know how to obtain t-distribution but I wonder if the above can be written as one since the terms in the numberator and denominator may not be independent.

Answer 1

The answer is $0$ because the variable you are considering is symmetric. [ You can write the expectation as an integral w.r.t. the joint distribution of the $X_i$'s. Since this joint distribution is symmetric we see that the required expectation is $0$].