Solutions to MAST 30020 Probability and Statistical Inference
نویسنده
چکیده
i=1 Ai \⋃ k 6=i Ak , {∑10 j=1 1Aj = 1} . (b) Setting A1 := B, A2 = A \ B and A3 = A, we see that Aj, j = 1, 2, 3, form a partition of Ω. The σ-algebra generated by the partition consists of all possible unions of its elements: σ({A,B}) = σ({A1, A3, A3}) = {∅, A1, A2, A3, A1 ∪ A2, A1 ∪ A3, A2 ∪ A3,Ω}. (c) No, F is not a σ-algebra, as the second axiom (A.2) fails: (B1 ×B2) doesn’t have the specified product form in the general case (take, for instance, Bj := [0, 1], j = 1, 2). 2. (a) Probability P is a set function F 7→ R given on a σ-algebra F on a set Ω such that (P.1) P(A) ≥ 0, A ∈ F , (P.2) P(Ω) = 1,
منابع مشابه
Accurate Inference for the Mean of the Poisson-Exponential Distribution
Although the random sum distribution has been well-studied in probability theory, inference for the mean of such distribution is very limited in the literature. In this paper, two approaches are proposed to obtain inference for the mean of the Poisson-Exponential distribution. Both proposed approaches require the log-likelihood function of the Poisson-Exponential distribution, but the exact for...
متن کاملMarginal Analysis of A Population-Based Genetic Association Study of Quantitative Traits with Incomplete Longitudinal Data
A common study to investigate gene-environment interaction is designed to be longitudinal and population-based. Data arising from longitudinal association studies often contain missing responses. Naive analysis without taking missingness into account may produce invalid inference, especially when the missing data mechanism depends on the response process. To address this issue in the ana...
متن کاملInformation Measures via Copula Functions
In applications of differential geometry to problems of parametric inference, the notion of divergence is often used to measure the separation between two parametric densities. Among them, in this paper, we will verify measures such as Kullback-Leibler information, J-divergence, Hellinger distance, -Divergence, … and so on. Properties and results related to distance between probability d...
متن کاملA Bayesian Approach to Protein Inference Problem in Shotgun Proteomics
The protein inference problem represents a major challenge in shotgun proteomics. In this article, we describe a novel Bayesian approach to address this challenge by incorporating the predicted peptide detectabilities as the prior probabilities of peptide identification. We propose a rigorious probabilistic model for protein inference and provide practical algoritmic solutions to this problem. ...
متن کاملComments on “A foundational justification for a weighted likelihood approach to inference”, by
Non-additive probability goes back to the very beginning of probability theory— the work of Jacob Bernoulli. Bernoulli’s calculus for combining arguments allowed both sides of a question to attain only small or zero probability, and he also thought the probabilities for two sides might sometimes add to more than one (Shafer 1978). Twentieth-century non-additive probability has roots in both mat...
متن کامل