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Question
If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P (X ≤ 5).
Solution
\[\text{ Mean} (np) = 3 \text{ and variance } (npq) = \frac{3}{2}\]
\[ \therefore q = \frac{1}{2}\]
\[\text{ and } p = 1 - \frac{1}{2}\]
\[n = \frac{Mean}{p}\]
\[ \Rightarrow n = 6\]
\[\text{ Hence, the distribution is given by } \]
\[P(X = r) = ^{6}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{6 - r} , r = 0, 1, 2 . . . 6\]
\[ = \frac{^{6}{}{C}_r}{2^6} \]
\[ \therefore P(X \leq 5) = 1 - P(X = 6) \]
\[ = 1 - \frac{1}{64}\]
\[ = \frac{63}{64}\]
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