Question

A certain plant dies at the end of the summer, producing seeds which might survive to the next summer. Suppose that N, the number of plants which result next summer, has a Poisson distribution with the parameter Î». Suppose that each of these plants goes through the same procedure the following year. If we let X_i denote the number of plants resulting the second summer from the ith plant of the next summer, then we assume N, X_1, X_2,... are independent and each has the Poisson distribution with parameter Î». Start with a single plant this summer. Let Y be the number of plants two summers from now (Y=X_1+X_2+...+X_N). 1) What is the probability generating function for Y, G_Y(s)? 2) What is P(Y=0)? What is P(Y=1)? 3) What is E(Y)?

          A certain plant dies at the end of the summer, producing seeds which might survive to the next summer. Suppose that N, the number of plants which result next summer, has a Poisson distribution with the parameter Î». Suppose that each of these plants goes through the same procedure the following year. If we let X_i denote the number of plants resulting the second summer from the ith plant of the next summer, then we assume N, X_1, X_2,... are independent and each has the Poisson distribution with parameter Î». Start with a single plant this summer. Let Y be the number of plants two summers from now (Y=X_1+X_2+...+X_N).

1) What is the probability generating function for Y, G_Y(s)?
2) What is P(Y=0)? What is P(Y=1)?
3) What is E(Y)?

Added by Jordan M.

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