Question:
In a binomial distribution, the mean is $4$ and variance is $3$. Then its mode is :
In a binomial distribution, the mean is $4$ and variance is $3$. Then its mode is :
Updated On: Jun 18, 2022
- 5
- 6
- 4
- None of these
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The Correct Option is C
Solution and Explanation
In Binomial distribution, Mean $= np$,
Variance $= npq$ and the mode is $r$
if for $x = r$, the probability function $p(x)$ is maximum.
Given $np = 4$ and $ npq = 3 $
$\therefore q = \frac{3}{4} p = 1 -q = 1 - \frac{3}{4} = \frac{1}{4} $
Also, $ n = \frac{4}{p} = \frac{4}{1/4} = 16 $
Now , $ \left(n+1\right)p = \left(16 +1\right) \frac{1}{4} = \frac{17}{4} = 4 + \frac{1}{4} $
$\Rightarrow $ The distribution will have unique mode (unimodal) & the mode $= 4$
Variance $= npq$ and the mode is $r$
if for $x = r$, the probability function $p(x)$ is maximum.
Given $np = 4$ and $ npq = 3 $
$\therefore q = \frac{3}{4} p = 1 -q = 1 - \frac{3}{4} = \frac{1}{4} $
Also, $ n = \frac{4}{p} = \frac{4}{1/4} = 16 $
Now , $ \left(n+1\right)p = \left(16 +1\right) \frac{1}{4} = \frac{17}{4} = 4 + \frac{1}{4} $
$\Rightarrow $ The distribution will have unique mode (unimodal) & the mode $= 4$
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Concepts Used:
Binomial Distribution
A common probability distribution that models the probability of obtaining one of two outcomes under a given number of parameters is called the binomial distribution. It summarizes the number of trials when each trial has the same probability of attaining one specific outcome. The value of a binomial is acquired by multiplying the number of independent trials by the successes.
Criteria of Binomial Distribution:
Binomial distribution models the probability of happening an event when specific criteria are met. In order to use the binomial probability formula, the binomial distribution involves the following rules that must be present in the process:
- Fixed trials
- Independent trials
- Fixed probability of success
- Two mutually exclusive outcomes