(note that captioned pmf is amended as question)
1. Explain why P(X is odd) = qP(X is even) and hence or otherwise show that P(X is odd) = q / (1+ q)
2. The random variable Y is independent of X and has pmf p(y)= qp^ y, y=1,2,...
Write down P(X=k and Y=k) for an arbitrary non-negative integer k and hence show that
P(X=Y)= p(1-p)/ [1-p(1-p)]
Using calculus, find the value of p that maximizes this expression and hence deduce the maximum possible value of P(X=Y) is 1/3.
(Note that you are not required to show any turning point that you locate is a maximum)
(I look into for a long while but have no idea yet.....please kindly assist if anyone expert is familiar with...Thanks a lot!)
更新1:
it is very appreciated and thanks for your prompt answer, myisland8132. But I can't get the MAX∣x=0 for x=p(1-p) as 1/4 by equating the f(x)=0. Would you kindly elaborate? Thanks a lot!
更新2:
sorry the value i can't deduce is the largest value for x=p(1-p) by equating the f'(x)=0 as 1/4. Please ignore the previous description. Thanks!