<Dl> <Dd> Σ = ∑ i = 1 N w i (∑ i = 1 N w i) 2 − ∑ i = 1 N w i 2 ∑ i = 1 N w i (x i − μ ∗) T (x i − μ ∗) = ∑ i = 1 N w i (x i − μ ∗) T (x i − μ ∗) V 1 − (V 2 / V 1). (\ displaystyle (\ begin (aligned) \ Sigma & = (\ frac (\ sum _ (i = 1) ^ (N) w_ (i)) (\ left (\ sum _ (i = 1) ^ (N) w_ (i) \ right) ^ (2) - \ sum _ (i = 1) ^ (N) w_ (i) ^ (2))) \ sum _ (i = 1) ^ (N) w_ (i) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) ^ (T) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) \ \ & = (\ frac (\ sum _ (i = 1) ^ (N) w_ (i) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) ^ (T) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right)) (V_ (1) - (V_ (2) / V_ (1)))). \ end (aligned))) </Dd> </Dl> <Dd> Σ = ∑ i = 1 N w i (∑ i = 1 N w i) 2 − ∑ i = 1 N w i 2 ∑ i = 1 N w i (x i − μ ∗) T (x i − μ ∗) = ∑ i = 1 N w i (x i − μ ∗) T (x i − μ ∗) V 1 − (V 2 / V 1). (\ displaystyle (\ begin (aligned) \ Sigma & = (\ frac (\ sum _ (i = 1) ^ (N) w_ (i)) (\ left (\ sum _ (i = 1) ^ (N) w_ (i) \ right) ^ (2) - \ sum _ (i = 1) ^ (N) w_ (i) ^ (2))) \ sum _ (i = 1) ^ (N) w_ (i) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) ^ (T) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) \ \ & = (\ frac (\ sum _ (i = 1) ^ (N) w_ (i) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right) ^ (T) \ left (\ mathbf (x) _ (i) - \ mu ^ (*) \ right)) (V_ (1) - (V_ (2) / V_ (1)))). \ end (aligned))) </Dd> <P> The reasoning here is the same as in the previous section . </P> <P> Since we are assuming the weights are normalized, then V 1 = 1 (\ displaystyle V_ (1) = 1) and this reduces to: </P>

What is weighted arithmetic mean discuss its advantages