<P> Extensions of this result can be made for more than two random variables, using the covariance matrix . </P> <P> In this case, one needs to consider </P> <Dl> <Dd> 1 2 π σ x σ y 1 − ρ 2 ∫∫ x y exp ⁡ (− 1 2 (1 − ρ 2) (x 2 σ x 2 + y 2 σ y 2 − 2 ρ x y σ x σ y)) δ (z − (x + y)) d x d y . (\ displaystyle (\ frac (1) (2 \ pi \ sigma _ (x) \ sigma _ (y) (\ sqrt (1 - \ rho ^ (2))))) \ iint _ (x \, y) \ exp \ left (- (\ frac (1) (2 (1 - \ rho ^ (2)))) \ left ((\ frac (x ^ (2)) (\ sigma _ (x) ^ (2))) + (\ frac (y ^ (2)) (\ sigma _ (y) ^ (2))) - (\ frac (2 \ rho xy) (\ sigma _ (x) \ sigma _ (y))) \ right) \ right) \ delta (z - (x + y)) \, \ mathrm (d) x \, \ mathrm (d) y .) </Dd> </Dl> <Dd> 1 2 π σ x σ y 1 − ρ 2 ∫∫ x y exp ⁡ (− 1 2 (1 − ρ 2) (x 2 σ x 2 + y 2 σ y 2 − 2 ρ x y σ x σ y)) δ (z − (x + y)) d x d y . (\ displaystyle (\ frac (1) (2 \ pi \ sigma _ (x) \ sigma _ (y) (\ sqrt (1 - \ rho ^ (2))))) \ iint _ (x \, y) \ exp \ left (- (\ frac (1) (2 (1 - \ rho ^ (2)))) \ left ((\ frac (x ^ (2)) (\ sigma _ (x) ^ (2))) + (\ frac (y ^ (2)) (\ sigma _ (y) ^ (2))) - (\ frac (2 \ rho xy) (\ sigma _ (x) \ sigma _ (y))) \ right) \ right) \ delta (z - (x + y)) \, \ mathrm (d) x \, \ mathrm (d) y .) </Dd>

Sum of squares of two gaussian random variables