Inequality regarding sum of squared probabilities

manpreet Best Answer 2 years ago

I'm working on a problem set for a course on Machine Learning and one the problems asks me to prove a given inequality. As an aid for that, the problem gives me the hint to use the following result, which I can understand but I am not able to prove:

Show that Σia2i≥a21+(1−a1)2C−1Σiai2≥a12+(1−a1)2C−1 for any CC real numbers such that a1≥a2≥...≥aC≥0a1≥a2≥...≥aC≥0 and Σiai=1Σiai=1

This is speacially useful in the exercise when the aiai are probabilities. However, I could not prove the result. I tried using contradictions and even induction but without any success. Any ideas on how to prove? Thanks a lot in advance!