If a random variable (x) follows binomial distribution with mean 5 and variance 4, and 5^{23}P(X=3)=\lambda 4^{\lambda}, then what is the value of \lambda?

Explanation

Mean np = 5 and variance npq = 4, so q = \frac{4}{5}, p = \frac{1}{5}, and n = 25. The probability P(X=3) = \binom{25}{3}(\frac{1}{5})^3(\frac{4}{5})^{22} = \frac{25 \times 24 \times 23}{6} \frac{4^{22}}{5^{25}} = 2300 \frac{4^{22}}{5^{25}} = 23(100) \frac{4^{22}}{5^{25}} = 23(25 \times 4) \frac{4^{22}}{5^{25}} = 23 \frac{4^{23}}{5^{23}}. So 5^{23}P(X=3) = 23(4^{23}). Comparing with \lambda 4^{\lambda}, we get \lambda = 23.

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