英国统计代考 Probability代写

STATISTICS

Probability: Theory and Applications

(Time allowed: 24 Hours)

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Considera Random Walk (RW) between neighbouring vertices of the infinite binary tree whose structure can be indicated pictorially, at least up to the first few levels, by the following picture:

If the RW is currently at vertex O, it is equally likely to move to either vertex C or vertex B at the next time step, independently of the past. Otherwise, if the RW is currently at some other vertex v then, independent of the past, on its next step it will move to vertex w with probability 1/3 if and only if v and w are neighbouring vertices. For example, from C the RW is equally likely to move to D, E or O at the next time step.

Let P_w represent the probability law of the RW when initially started at some vertexw.DefineT_w to be the time the RW first hits vertex w, where T_w = 0 if the RW

is initially started at vertex w and T_w = +∞ if the RW never hits vertex w at any non-negative time. As usual, define P(F ; G) := P(F ∩ G) for any events F and G.

The shrinking size of branches in the diagram is only to permit easy drawing; distances in the picture are not important for the RW, only the neighbourhood structure between vertices is significant. For example, by symmetry, P_B(T_O < ∞)

will have the same value as P_E(T_C < ∞). 英国统计代考

Show that

(a) P_C(T_O < ∞) = ¹ , [8]

(b) P_O(T_C < ∞) = ² , [4]

(d) P_C(T_D < ∞ ; T_E < ∞) = ¹⁸ . [4]

2.Considera discrete time branching process Z = (Z_n)_n_≥₀ with Z₀ = 1, where Z_n

represents the number of individuals in a population at time n. At time n + 1, each

individual in the n^th generation is replaced by a random number of new individuals, each distributed like an independent copy of some offspring random variable, X, taking values in {0, 1, 2, . . . } and satisfying µ := E(X) < ∞.

Define the probability generating function (PGF) of the offspring distribution X by

(a)Calculate E(θ^Zn+1 Z_n= k) in terms of g(θ) and k, hence deduce an expres- sion for E(θ^Zn+1 Z_n). Explain the fundamental difference between these two conditional [3]

(b)Definethe PGF for Z_n as g_n(θ) := E(θ^Zn ). Prove that g_n₊₁(θ) = g_n(g(θ)). 英国统计代考

Deduce an expression for g_n(θ) depending only on g, θ, and n. [2]

For the remainder of this question, consider the special case when the offspring random variable, X, satisfies

P(X = k) = (1/2)^k⁺¹ (k = 0, 1, 2, . . . ).

(c)Show that g(θ) = 1/(2 − θ) and that g^j(1)= [2]

(d)Writedown g₀(θ) and g₁(θ). Prove by induction that, for all n ∈ {0, 1, 2, . . . },

[5]

(e)Deduce that P(Z_n= 0) = n/(n + 1). Hence, prove that the population will eventually become extinct, almost [3]

(f)Byexpanding g_n(θ) as a power series in θ, determine P(Z_n = k) for k ≥ [5]

3.Let Z be a Poisson random variable (RV) of parameter µ > 0,written Z P o(µ), so that 英国统计代考

Suppose X₀, X₁, . . . is a sequence of independent random variables such that, for each i ≥ 0, X_i is a Poisson RV of parameter λ_i, where

µ .2 Σ_i

(a)Calculatethe probability generating function (PGF) of Z,

where θ ∈ [0, 1]. [2]

(b)Writedown the PGF of X_i, g_i(θ) := E(θ^Xi ). [1]

(c)Define Y_n:= Σn X_i. Find the PGF of Y_n, h_n(θ) := E(θ^Yn ). Hence, deduce

that Y_n is a Poisson RV with rate parameter µ(1 − (2/3)ⁿ⁺¹). Hint: Recall that 1 + x + x² + · · · + xⁿ = (1 − xⁿ⁺¹)/(1 − x) for x ƒ= 1. [3] 英国统计代考

(d)Forθ ∈ [0, 1], show that h_n(θ) → e^µ⁽^θ⁻¹⁾ as n → ∞. Stating any general result that you appeal to, deduce that Y_n converges in distribution to Z, that is,

for each k ∈ {0, 1, 2, . . . }. [2]

(e)DefineY := ^∞i=0 X_i. Use continuity of probability (or monotone convergence) to prove that P(Y_n ≤ k) → P(Y ≤ k). Deduce that Y ∼ P o(µ). [2]

(f)Deducethat Y Y_n = ^∞i=n+1 X_i P o(γ_n) for some parameter γ_n that you should calculate and Hence, show that 英国统计代考

(i) E(|Y − Y_n|) → 0 as n → ∞,

(ii) Y_n → Y in probability. [6]

(g)Stating any general result to which you appeal, show that, with probability one,only finitely many of the events {X₀ ≥ 1}, {X₁ ≥ 1}, {X₂ ≥ 1}, . . .

Deduce that Y_n → Y almost surely. [4]

4.Recallthat a Poisson point process on G ⊂ Rn with intensity function λ : G → 英国统计代考

[0, ∞) is a random collection of points in G such that:

(P1) if G₁, G₂, . . . are disjoint subsets of G, then the numbers N (G₁), N (G₂), . . .

of points occurring in G₁, G₂, . . . are independent random variables;

(P2) N (G_k) has the Poisson distribution of parameter x∈ttk λ(x)dx.

(a)Letν, µ > An experiment is performed where Red and Blue particles hit a sensor at random. Suppose that the positions where Red particle hit the sensor form a Poisson point process on R2 of constant intensity ν. Suppose that the positions where Blue particle hit the sensor form a Poisson point process on R2 of constant intensity µ. In addition, suppose that the positions where Red and Blue particles hit are independent.

i.Describe,with reasons, the combined process for the positions of all par- ticles (ie. regardless of whether Red or Blue) hitting the [2]

ii.Findthe distribution of the distance, Z, from the origin to the closest particle. [2]

iii.What’s the probability the two particles that are closest to the origin consistof one of each colour? [3] 英国统计代考

iv.Given that exactly four Red particles hit the sensor inside the square shaped region S = [ 2, +2]², explain why U has conditional probability densityfunction

where U := inf{x > 0 : at least one Red particle hit within [−x, x]2}.

Hint: You should recall and use the ‘conditional uniformity’ principle. [4]

(b) Let u, v > 0. Suppose that N is a Poisson point process on (0,∞) with

intensity function λ(s) = 1/(s(1 + s)). 英国统计代考

(i) Show that the probability no points of N occur in the interval (u, v] is u

Hint: Use partial fractions and recall that (ln x)0 = 1/x. [4]

(ii) Write down the probability distribution and mean for the number of points

of N occurring in the interval (u, v].[2]

(iii) Do fifinitely or infifinitely many points occur in the interval (u, ∞)? Brieflfly

explain your answer.[3]

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