Question:

Find the error in the following argument.

\(X\tilde{\ }Poisson\left( \lambda \right)\). Recall that \({{\mu }_{X}}=\sigma _{X}^{2}=\lambda \).

Let Y = 2X.

1. Since \(\operatorname{var}\left( \alpha X \right)={{\alpha }^{2}}\operatorname{var}\left( X \right)\), \(\sigma _{Y}^{2}={{2}^{2}}\sigma _{X}^{2}=4\lambda \).
2. Y = 2X = X + X. Since \(X\tilde{\ }Poisson\left( \lambda \right)\) and Y is the sum of two Poisson random variables with parameter \(\lambda \), \(Y\tilde{\ }Poisson\left( 2\lambda \right)\). Therefore \(\sigma _{Y}^{2}=2\lambda \).

The two calculations of \(\sigma _{Y}^{2}\) give two different values, leading to a contradiction.

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