Let X ∼ Poisson(λ) and Y ∼ U[X, 2X]. Find E(Y ) and V ar(Y ).

Solutions

Expert Solution

For a poisson distribution, the mean is equal to its variance which is equal to its parameter. Therefore, we have here:

Now, the conditional distribution of Y given X here is obtained as:

For compound distribution Y, the expected value and variance here are obtained as:

This is the required expected value of Y here.

Now the variance of Y here is obtained as:

The second moment of X here is obtained as:
E(X²) = Var(X) + [E(X)]² =

Therefore, we get the variance of Y here as:

This is the required variance of Y here.

orchestra answered 1 year ago

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