MAFS5020 Advanced Probability and Statistics
Assignment 1
高级概率与统计代写 Problem 1 (Required, 30 marks) (a) We let Ω = {ω1, ω2, ω3, ω4, ω5, ω6 } be a sample space and let ω1 = {ω1, ω2 } and ω2 = {ω3, ω4 } be two events
Problem 1 (Required, 30 marks) 高级概率与统计代写
(a) We let Ω = {ω1, ω2, ω3, ω4, ω5, ω6 } be a sample space and let ω1 = {ω1, ω2 } and ω2 = {ω3, ω4 } be two events
(i) Write down the smallest σ-algebra that contains E1, E2 (denoted by σ(E1, E2 ). No reason is required.
(ii) Define ℱ = σ(E1, E2 ) ∪ {{ω5, ω6 }}, determine if ℱ is σ-algebra. Explain your answer.
(b) We let Ω be a sample space and let ℱ1 and ℱ2 be two σ-algebra on Ω.
Prove that the set defined by
ℱ1 ∩ ℱ2 = {A ⊆ Ω：A ∈ ℱ1 and A ∈ ℱ2 }
is a -algebra.
(☺Hint: You just need to verify that the set satisfies the three conditions stated in the definition of σ-algebra.)
Problem 2 (Required, 25 marks) 高级概率与统计代写
We consider a motor vehicle insurance issued by an insurance company. The insurance company predicts that
- The number of claims made by the policy in a year, denoted by N, has Poisson distribution with mean λ.
- The insurance company does not know the true value of λand conjectures that λhas the following distribution:
P (λ=1) =0.2, P (λ=2) =0.6, P (λ=3) =0.2.
After two years, the insurance company is going to estimate the distribution of N using the data observed in the past two years.
(a) Suppose that no claims are made by policyholder in the first year and 3 claims has made in second year, calculate the probability that there is at least 1 claim made by the policyholder in the coming year.
(b) Suppose that the policyholder made a total of 3 claims in the past two years, calculate the probability that there is at least 1 claim made by the policyholder in the coming year.
Problem 4 (Optional, 30 marks) 高级概率与统计代写
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 Fx(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 and give formula for each probability.
(a) P(a ＜ X ≤ b) and P(a ≤ X ≤ b) for any real and 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.)
