Rachunek Prawdopodobieństwa 2

Rachunek Prawdopodobieństwa 2

Zestaw zadań nr 3

Termin realizacji: 31 X 2008

1. Pokazać, że G X (e ^t ) = P

n t

n1 E(X ⁿ ).

2. Let X ≥ 0 have a probability generating function G and write t(n) = P (X > n). (i) Show that the generating function of the sequence t(n), n ≥ 0, is T (s) = (1 − G(s))/(1 − s). (ii) Show that EX = T (1).

3. Let X have the binomial distribution Bin(n, U ), where U is uniform on (0, 1). Show that X is uniformly distributed on {0, 1, 2, . . . , n}.

4. Let X have the Poisson distribution with parameter Y , where Y has the Poisson distribution with parameter µ. Show that G X+Y (x) = exp{µ(xe ^x−1 − 1)}.

5. Let X have the Poisson distribution with parameter Λ, where Λ is exponential with parameter µ. Show that X has a geometric distribution.

6. Udowodnić, że (1 − 4pqs ² ) ^−1/2 jest funkcją tworzącą ciągu p 0 (n) = P (S n = 0).

7. Use generating functions to show that (i) 2kf 0 (2k) = P (S 2k−2 = 0) for k ≥ 1, and (ii) P (S 1 · · · S 2n 6= 0) = P (S 2n = 0) for n ≥ 1.

8. A particle performs a random walks on the corners of the square ABCD, where the 1-step transition probabilities are ρ AB = ρ BA = ρ CD = ρ DC = α and ρ AD = ρ DA = ρ CB = ρ BC = β, and α + β = 1. Let G A (s) be the generating function of the sequence p AA (n), n ≥ 0, where p AA (n) is the probability that the particle is at A after n steps, having started from A. (i) Compute G A (s). (ii) Find the generating function of the time of the first return to A.

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