Maximum Value of Binomial Random VariablesDate: 09/12/2001 at 12:03:15 From: Clark Thorne Subject: Maximum Value of Binomial Random Variables For a binomial random variable X with parameters (n,p), show that P{X=i} first increases and then decreases, reaching its maximum value when i is the largest integer less than or equal to (n+1)p. I first tried to differentiate the binomial distribution, but fell on my face with the gamma function. This seems intuitive, but how do you show this for generic parameters (n,p)? Date: 09/12/2001 at 15:38:00 From: Doctor Jubal Subject: Re: Maximum Value of Binomial Random Variables Hi Clark, Thanks for writing to Dr. Math. Based on your question, I'll assume you know that for a binomial distribution, the probability that X=i is given by n! P{X=i} = p^i * (1-p)^(n-i) * ---------- i!(n-i)! where n is the number of trials and p is the probability of some outcome on a given trial. I assume you tried to differentiate this, and found it hard. Let me suggest a different way. Let's compare the probability that X=i to the probability that X=i+1. If P{X=i}/P(X=i+1) < 1, then the distribution function is increasing. If the ratio is greater than one, the distribution function is decreasing. For the binomial distribution n! i (n-i) ---------- P{X=i} p * (1-p) * i!(n-i)! ---------- = ----------------------------------------- P{X=i+1} (i+1) (n-i-1) n! p * (1-p) * ---------------- (i+1)!(n-i-1)! By canceling terms that appear in both numerator and denominator, we can simplify this to P{X=i} (1-p) * (i+1)!(n-i-1)! (1-p)(i+1) ---------- = ------------------------ = ------------ P{X=i+1} p * i!(n-i)! p(n-i) Since the distribution function is increasing when this ratio is less than one, the binomial function is increasing when p(n-i) > (1-p)(i+1) np - ip > i + 1 - ip - p np > i + 1 - p np - p > i + 1 (n+1)p > i + 1 Similarly, the binomial function is decreasing when i + 1 > (n+1)p. This is exactly what you were trying to prove. So long as i+1 is less than (n+1), P{X=i+1} is greater than P{X=i}, and the distribution function is increasing. But if i is the largest integer less than or equal to (n+1)p, then i+1 is greater than (n+1)p, and the distribution function is decreasing for this and all greater values of i. Therefore, the binomial function is increasing for small i, decreasing for large i, and has a single maximum at i = floor[ (n+1)p ]. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jubal, The Math Forum http://mathforum.org/dr.math/ |
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