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Question
Find the binomial distribution whose mean is 5 and variance \[\frac{10}{3} .\]
Solution
Mean of binomial distribution, i.e. np =5
Variance, i.e. npq = \[\frac{10}{3} \]
\[q = \frac{\text{ Variance } }{\text{ Mean } } = \frac{2}{3}\]
\[\text{ and } p = 1 - q = \frac{1}{3}\]
\[np = 5 \]
\[ \Rightarrow n = 15\]
\[ \therefore P(X = r) = ^{15}{}{C}_r \left( \frac{1}{3} \right)^r \left( \frac{2}{3} \right)^{15 - r} , r = 0, 1, 2 . . . . . . 15\]
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