If a random variable \(X\) has a Poisson distribution with mean 5, then the…
2017
If a random variable \(X\) has a Poisson distribution with mean 5, then the expectation \(E\left [ \left ( x+2 \right )^{2} \right ]\) equals ___.
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Correct answer: 54
We want to compute E[(X+2)^2] for X having a Poisson distribution with mean 5.
Use the identity: E[(X+2)^2] = E[X^2] + 4E[X] + 4.
For a Poisson(5) distribution, E[X] = 5.
Variance of X is 5, so E[X^2] = Var(X) + (E[X])^2 = 5 + 25 = 30.
Now substitute: E[(X+2)^2] = 30 + 4*5 + 4 = 30 + 20 + 4 = 54.
Final answer: 54