# fixed points – Properties of a function based on two iid Binomial random variables

Let $$k in {2,3,4,dots}$$ and $$0 Suppose that $$X$$ and $$Y$$ are two independent and identically distributed Binomial random variables with parameters $$k$$ and $$p.$$ Let $$Z = 2X-Y.$$ Consider the function $$h_k(p) = P(Z < k).$$ Show the following:

a) The function $$h_k(p)-p$$ is decreasing in $$p.$$

b) $$h_k(p)$$ has a unique fixed point $$p_k^{ast}$$ such that $$p_k^{ast} > frac{1}{2}.$$

c) $$|h_k'(p_k^{ast})| < 1.$$

I should probably add that the above three statements are conjectures based on numerical computations for different values of $$k.$$ I would be very grateful if someone can help me with providing analytic proofs of the above statements (if they are true) or a counterexample in case any of the above statement is incorrect.