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if x is a poisson variate such that p(x=1)= p(x=2) find e(x^2)

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Given that P(X=1) = P(X=2), we can write:

\begin{aligned} P(X=1) &= e^{-\lambda}\frac{\lambda^1}{1!} \\ P(X=2) &= e^{-\lambda}\frac{\lambda^2}{2!} \end{aligned}

Dividing the two equations, we get:

\begin{aligned} \frac{P(X=2)}{P(X=1)} &= \frac{e^{-\lambda}\frac{\lambda^2}{2!}}{e^{-\lambda}\frac{\lambda^1}{1!}} \\ &= \frac{\lambda}{2} \end{aligned}

Since P(X=1) = P(X=2) , we have:

\begin{aligned} \frac{P(X=2)}{P(X=1)} &= 1 \\ \frac{\lambda}{2} &= 1 \end{aligned}

Therefore,  \lambda = 2. Now, we need to find E(X^2):

\begin{aligned} E(X^2) &= \sum_{x=0}^\infty x^2 P(X=x) \\ &= \sum_{x=0}^\infty x^2 \frac{e^{-\lambda}\lambda^x}{x!} \\ &= \lambda^2 + \lambda \\ &= 2^2 + 2 \\ &= 6 \end{aligned}

So, E(X^2) = 6.

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