Binomial Distribution
Binomial distribution is the distribution of a total number of successes in a given number of Bernoulli trials. The common notation is
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For a fixed number of trials n, the binomial distribution always behaves in the same way: as a function of k, it is monotone increasing up to a certain point m after which (perhaps with an exception of the next point) it is monotone decreasing.
Indeed,
b(k; n, p) / b(k1; n, p)  = C(n, k) p^{k}(1  p)^{n  k} / C(n, k1) p^{k1}(1  p)^{n  k + 1}  
(1)  = (n  k + 1) p / k(1  p)  
= 1 + ((n + 1) p  k) / k(1  p).  
It is clear now that the right hand side in (1) is greater than 1 whenever
b(m; n, p) = b(m  1; n, p).  
In any event, there is only one integer m that satisfies
(2)  (n + 1) p  1 < m ≤ (n + 1) p. 
Summing up, as a function of k, the expression b(k; n, p) is monotone increasing for
The number m that satisfies (2) is known as the most probable (most likely) number of successes in n Bernoulli trials. As a matter of fact,
∑ k b(k; n, p)  = ∑ k C(n, k) p^{k}q^{n  k}  
= pq^{n1}∑ k C(n, k) (p/q)^{k  1}  
= pq^{n1} n (1 + p/q)^{n  1}  
= pq^{n1} n (q + p)^{n  1} / q^{n  1}  
= p n,  
where we used the identity
n (1 + x)^{n  1}  = ∑ k C(n, k) x^{k  1}.  
The result for the expected value np = ∑ k b(k; n, p) might have been anticipated given the interpretation of the probability as a relative frequency.
Note that the expected value np is always different from the most likely value

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