Proof for Bhattacharyya Coefficient

[Manasa]

Can someone explain how the square roots of the probabilities are considered as the direction cosines of the two vectors.

 

[Dr.Peterson]

What is the definition of direction cosines?

Are you aware that an essential characteristic of them is that the sum of their squares is 1? See equation (4) in http://mathworld.wolfram.com/DirectionCosine.html.

 

[Manasa]

Yes I'm aware of. But how can we get the square root of p1, p2,... as direction cosine. Would you elaborate please? some image of solution would help me a lot. Thanks.

 

[Dr.Peterson]

As far as I can tell (knowing nothing of the context), all they are saying is that [MATH]\left(\sqrt{p_1},\sqrt{p_2},\dots\sqrt{p_k}\right)[/MATH] and [MATH]\left(\sqrt{p'_1},\sqrt{p'_2},\dots\sqrt{p'_k}\right)[/MATH] can be thought of as two unit vectors in k-space -- just as they are. I don't think there's anything to be done to them (at this point).