IGNOU MPH 11 2026 - Show under what condition Poisson distribution tends to Normal distrib

Question

Show under what condition Poisson distribution tends to Normal distribution.

19 Jan 2026

Answer :

Word Count : 325

The Poisson distribution is defined as: [ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots ] where (\lambda) is the mean and variance of the distribution. The mean and variance are equal: [ \mu = \sigma^2 = \lambda ] To see when the Poisson distribution approaches the Normal distribution, consider the limit of large (\lambda). For large (\lambda), (k!) can be approximated using Stirling’s approximation: [ k! \approx \sqrt{2 \pi k} \left(\frac{k}{e}\right)^k ] Substituting into the Poisson probability: [ P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!} \approx \frac{\lambda^k e^{-\lambda}}{\sqrt{2

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