If a random variable (x) follows binomial distribution with mean 5 and variance 4 and 5²³ P(X=3) = λ4^λ, then what is the value of λ?

Mean np = 5, variance npq = 4. So q = 4/5, p = 1/5, n = 25. P(X=3) = C(25,3) p³ q²² = C(25,3) (1/5)³ (4/5)²² = C(25,3) · 4²² / 5²⁵. So 5²⁵ P(X=3) = C(25,3) · 4²² = 2300 · 4²². Then 5²³ P(X=3) = 5²⁵ P(X=3)/25 = 2300 · 4²²/25 = 92 · 4²². Hmm, this should equal λ · 4^λ. With λ = 23: 23 · 4²³ = 23 · 4 · 4²² = 92 · 4²². Match! So λ = 23.