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Question
If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.
Solution
\[\text{ For binomial distribution of X } , \]
\[P(X = r) = ^{n}{}{C}_r (p )^r (q )^{n - r} , r = 0, 1, 2, . . . , n\]
\[P(X = 1) = np(q )^{n - 1} \]
\[P(X = 2) =^{n}{}{C}_2 p^2 (q )^{n - 2} \]
\[ \Rightarrow np(q )^{n - 1} = ^{n}{}{C}_2 p^2 (q )^{n - 2} = \alpha \]
\[\text{ Simplifying the above equation we get,} \]
\[q = \frac{n - 1}{2}p\]
\[ \Rightarrow 2q = np - p \]
\[\text{ On putting, q = 1 - p we get } \]
\[2 - 2p = np - p \]
\[p(n + 1) = 2 . . . . . (i)\]
\[\text{ Also} , P(X = 1) = \alpha\]
\[ \Rightarrow np(1 - p )^{n - 1} = \alpha . . . . . (ii)\]
Note: We cannot find the value of n as (i) and (ii) are not linear and hence we cannot find the value of P(X = 4)
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