Math Problem Statement

Recall the Taylor expansion of ex = 1 + x+ x2/2! + x3/3! +... (a) Find the Taylor expansion of eλ,e−λ. (b) Find the Taylor expansion for eλ + e−λ. (c) Assume that X∼Possion(λ), Find the probability that X is even.

Solution

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Math Problem Analysis

Mathematical Concepts

Taylor Series
Exponential Functions
Probability Theory
Poisson Distribution

Formulas

e^x = 1 + x + x^2/2! + x^3/3! + ...
e^λ = 1 + λ + λ^2/2! + λ^3/3! + ...
e^−λ = 1 - λ + λ^2/2! - λ^3/3! + ...
P(X = k) = λ^k * e^−λ / k!
P(X is even) = (1 + e^−2λ) / 2

Theorems

Taylor Series Expansion
Properties of Exponential Functions
Poisson Probability Theorem

Suitable Grade Level

Undergraduate Mathematics