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Generating Function of Transformed Nonnegative Integer-Valued Random Variable

This study explores the properties of generating functions in relation to nonnegative integer-valued random variables. Specifically, we define the generating function ( G_X(t) ) for a random variable ( X ), and derive the generating function ( G_Y(t) ) for the transformed variable ( Y = aX + b ), where ( a ) and ( b ) are nonnegative integers. The results show that the generating function of ( Y ) is ( G_Y(t) = t^b G_X(t^a) ), affirming that ( Y geq b ) with implications for probability distributions.

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Generating Function of Transformed Nonnegative Integer-Valued Random Variable

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  1. 23. Let GX(t) be the pgf of a nonnegative integer-valued random variable X. Let Y = aX + b, where a; b are nonnegative integer. Show that GY(t) = tbGX(ta). It is clear that Y b.

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