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Finding Mode of Binomial Distribution

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The mode of the binomial distribution is the value of X which maximizes the value of the binomial probability mass function. Thus if X = r gives mode then we should have p(r) > p(r – 1) \text{ and } p(r) > p(r + 1).

How to find the Mode of the Binomial Distribution?

Let X be a Binomial variate with parameters n and p.

Case 1: When (n+1)p is an integer

Let (n + 1) p = k (an integer).
In this case, the distribution is bi-modal, the two modal values being X=k and X=k–1.

Example: Suppose n = 9 and p = 0·4, then (n + 1) p = 10 × 0·4 = 4, which is an integer.

Hence, in this case, the distribution is bi-modal, the two modal values being 4 and 4 – 1 = 3.

Case 2: When (n+1)p is not an integer

Let (n + 1)p = k + f, where k is the integral part and f is the fractional part of (n + 1)p. In this case, the distribution has a unique mode at X = k, the integral part of (n + 1) p.

Example: If n = 7 and p = 0·6, then (n + 1) p = 8 × 0·6 = 4·8.

Hence the distribution is unimodal and Mode = 4, the integral part of 4·8.

Proof of Mode of Binomial Distribution:

We want to find where the maximum value of the probability distribution function is attained. The probability distribution function of the binomial distribution is given as, P(X=x) = {n\choose x}p^xq^{n-x}

We first calculate the ratio p(x)/p(x-1). We then divide our proof into two cases. We first deal with the case when (n+1)p is not an integer.

Mode of Binomial Distribution Proof Case 1

We now deal with the case where (n+1)p is not an integer. We have two modes in this case.

Mode of Binomial Distribution Proof Case 2

How many modes does the Binomial Distribution have?

The binomial distribution is either unimodal or bimodal depending on the value of the parameters of the distribution.

If (n+1)p is an integer then the binomial distribution has two modes equal to X=(n+1)p and X=(n+1)p+1.

If (n+1)p is not an integer then the binomial distribution has a unique mode equal to the greatest integer less than (n+1)p.

Is Mean and Mode equal for Binomial Distribution?

If np is a whole number (i.e., integer), then the distribution is unimodal and the mean and mode are equal each being np.

What is the Mode in Bernoulli Distribution?

If the value of p is equal to 0.5, then the Bernoulli distribution is bimodal and the values of the mode are equal to 0 and 1.

If the value of p is not equal to 0.5 then the Bernoulli distribution is unimodal and the mode is the value of X corresponding to the value greatest among p and (1-p).

References:

Business Statistics – SC Gupta and Indra Gupta.

Fundamentals of Mathematical Statistics – SC Gupta

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