高级概率代写 MAFS5020代写 高级概率和统计代写 基本概率论代写

MAFS5020 Advanced Probability and Statistics

Problem Set 1

Topic 1: Basic Probability Theory

高级概率代写 Problem 1 We let Ω = {ω1, ω2, ω3, ω4, ω5, ω6} be a sample space with 6 outcomes. Define ℱ = {Ω, Φ , A, B, C, Ac, Bc, Cc }, where A= {ω1, ω2}, 

Problem 1

We let Ω = {ω₁, ω₂, ω₃, ω₄, ω₅, ω₆} be a sample space with 6 outcomes. Define ℱ = {Ω, Φ , A, B, C, A^c, B^c, C^c }, where A= {ω₁, ω₂}, B={ω₃, ω₄} and C= {ω₅, ω₆}.

(a) Show that ℱ is a σ-algebra.

(b) Show that σ(A,B)=ℱ , where σ(A,B) denotes the smallest σ-algebra containing the events A and B.

Problem 2 高级概率代写

Given a sample space Ω, we let ℱ₁ and ℱ₂ be two σ-algebra in Ω.

(a) We define ℱ₁ ∩ ℱ₂ = {A ⊆ Ω：A ∈ ℱ₁ and B ∈ ℱ₂ }. Determine if ℱ₁ ∩ ℱ₂ is a σ- algebra.

Problem 3 高级概率代写

(a) We let Ω = {ω₁, ω₂, ω₃} be a sample space and let σ(ω₁) be the smallest σ- algebra containing the set {ω₁}.

Define ℱ = σ(ω₁) ∪ {ω₂}, Determine if ℱ is a σ-algebra.

(b) We let Ω be a sample space and let ℱ₁ and ℱ₂ be two σ-algebra on Ω. Show that the set defined by

ℱ₁ ∩ ℱ₂ = {A ⊆ Ω：A ∈ ℱ₁ and A ∈ ℱ₂ }  

is a σ-algebra.

Problem 4  高级概率代写

We let M,N be two independent Poisson random variables with parameters λ_M and λ_N respectively. For any positive integer n, k, compute the probability P(M= k∣M+N=n) .

Problem 5

We let X and Y be non-negative, discrete random variables with probability distributions:

Problem 6 (Harder) 高级概率代写

In lecture note, we say a random variable is measurable if and only if X⁻¹(−∞ , X］= {ω：X(ω) ≤ x } ∈ ℱ for all x ∈ R so that F_X(x) is well-defined.

In fact, there are some other possible ways to define a measureable random variable.

We let X be a random variable defined on a probability space ( Ω, ℱ, P ), we say X is measureable if and only if

X⁻¹(a,b)= {ω:a ﹤X(ω) ﹤ b } ∈ ℱ,

for all real a﹤b . (i.e. the probability P(a﹤X﹤b) is well-defined).

Suppose that X is measurable according to this new definition, show that the following probabilities are well-defined:

(a) P(a﹤X≤b) and  P(a≤X≤b) for any real a and b with a﹤b .

(b) P(X≤x ) for any x ∈ R  .

(Hint: To prove (b), you may consider the set {ω:-n≤X≤x } for any positive integer n and use Monotone convergence theorem.)

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