Let X be an ensemble of M points a1，…，aM and Let PX（aM)=α，Show that H（X)=αlog（1/α)+（1

Let X be an ensemble of M points a1，…，aM and Let PX(aM)=α，Show that H(X)=αlog(1/α)+(1 - α)log(1/(1 - α))+(1 - α)H(Y) where Y is an ensemble of M-1 points a1，…，aM-1 with probabilities PY(aj)=PX(aj)／(1 - α)；1≤j≤M - 1．Show that H(X)≤αlog(1/α)+(1 - α)log(1/(1 - α))+(1 - α)log(M - 1) and determine the condition for equality．