Question: Suppose that Y is discrete-valued, taking values only on the non-negative integers, and the conditional distribution of Y given X x is Poisson: P

Suppose that Y is discrete-valued, taking values only on the non-negative integers, and the conditional distribution of Y given X Æ x is Poisson:

P

£

Y Æ j j X Æ x

¤

Æ

exp

¡

¡x0¯

¢ ¡

x0¯

¢j j !

, j Æ 0, 1, 2, ...

Compute E[Y j X] and var[Y j X] . Does this justify a linear regression model of the formY Æ X0¯Åe?

Hint: If P

£

Y Æ j

¤

Æ

exp(¡¸)¸j j !

then E[Y ] Æ ¸ and var[Y ] Æ ¸.

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