Let X be random variable with the binomial distribution with parameters n and 0 < p < 1.
(1) Show that (P(X = x) / P(X = x -1)) - 1 = np + (p - x)) / (x(1-p)) for any 1 ≤ x ≤ n.
(2) Show that when 0 ≤ x < (n + 1)p , P(X = x) is an increasing function x and for (n + 1)p < x ≤ n, P(X = x) is a decreasing function x.
(1)
(2)
For 0 ≤ x < (n + 1)p
(n + 1)p - x > 0
then,
(P(X = x) / P(X = x -1)) - 1 = np + (p - x)) / (x(1-p)) = ((n + 1)p - x ) / (x(1-p)) > 0
=> (P(X = x) / P(X = x -1)) > 1
=> P(X = x) > P(X = x -1)
=> P(X = x) is an increasing function of x
For (n + 1)p < x ≤ n
=> (n + 1)p - x < 0
(P(X = x) / P(X = x -1)) - 1 = np + (p - x)) / (x(1-p)) = ((n + 1)p - x ) / (x(1-p)) < 0
=> (P(X = x) / P(X = x -1)) < 1
=> P(X = x) < P(X = x -1)
=> P(X = x) is an decreasing function of x
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