Math Problem Statement

The extracted question from the user-uploaded image includes multiple mathematical problems, including Bayes' theorem, probability, continuous random variables, and harmonic functions.

Solution

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Math Problem Analysis

Mathematical Concepts

Bayes' Theorem
Probability
Normal Distribution
Harmonic Functions
Continuous Random Variables

Formulas

Bayes' Theorem: P(A|B) = [P(B|A) * P(A)] / P(B)
Variance transformation: Var(Y) = a^2 * Var(X) when Y = aX + b
Binomial Probability Formula: P(X = k) = nCk * p^k * (1-p)^(n-k)

Theorems

Bayes' Theorem
Cauchy's Integral Formula

Suitable Grade Level

Undergraduate Mathematics

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