The mode of a discrete random variable X with pmf p(x) is that value x* for which p(x) is largest (the most probable x value).
a. Let X ∼Bin(n, p). By considering the ratio b( x +1; n , p) / b (x; n , p), show that b(x; n, p) increases with x as long as x < np – ( 1 – p). Conclude that the mode x* is the integer satisfying (n + 1)p - 1≤ x* ≤ (n + 1) p.
b. Show that if X has a Poisson distribution with parameter μ, the mode is the largest integer less than μ. If μ is
an integer, show that both μ – 1 and μ are modes.
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