If a random variable $X$ has a Poisson distribution with mean 5, then find the expectation $E[(X + 2)^2]$.

(GATE 2017)

Answer: 54

Explanation:

Here, $X$ is a random variable that follows a Poisson distribution with mean $E(X) = 5$.

Therefore, its variance $Var(X) = 5$.

Now, $Var(X) = E(X^2) - \big\{E(X)\big\}^2$

$\Rightarrow E(X^2) = 5 + 5^2$

$\Rightarrow E(X^2) = 30$

Therefore, $E[(X + 2)^2] $

$= E(X^2 + 4X + 4)$

$= E(X^2) + E(4X) + E(4)$

$= E(X^2) + 4E(X) + 4$

$= 30 + 4 \times 5 + 4$

$= 54$

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