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.

To address each part of the question:

### Part (a): Taylor Expansion of \( e^{\lambda} \) and \( e^{-\lambda} \)

The Taylor expansion of \( e^{x} \) around \( x = 0 \) is given by:
\[
e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots
\]

1. **For \( e^{\lambda} \):**
   Substitute \( x = \lambda \):
   \[
   e^{\lambda} = 1 + \lambda + \frac{\lambda^2}{2!} + \frac{\lambda^3}{3!} + \dots
   \]

2. **For \( e^{-\lambda} \):**
   Substitute \( x = -\lambda \):
   \[
   e^{-\lambda} = 1 - \lambda + \frac{\lambda^2}{2!} - \frac{\lambda^3}{3!} + \dots
   \]

### Part (b): Taylor Expansion of \( e^{\lambda} + e^{-\lambda} \)

To find the Taylor expansion of \( e^{\lambda} + e^{-\lambda} \), add the two expansions we derived in part (a):
\[
e^{\lambda} + e^{-\lambda} = \left(1 + \lambda + \frac{\lambda^2}{2!} + \frac{\lambda^3}{3!} + \dots\right) + \left(1 - \lambda + \frac{\lambda^2}{2!} - \frac{\lambda^3}{3!} + \dots\right)
\]

Notice that the terms with odd powers of \( \lambda \) will cancel out, while the even powers will remain. Thus:
\[
e^{\lambda} + e^{-\lambda} = 2 + \frac{2 \lambda^2}{2!} + \frac{2 \lambda^4}{4!} + \dots = 2 \left(1 + \frac{\lambda^2}{2!} + \frac{\lambda^4}{4!} + \dots \right)
\]
This series represents the Taylor expansion for \( e^{\lambda} + e^{-\lambda} \), which is equivalent to \( 2 \cosh(\lambda) \).

### Part (c): Probability that \( X \) is even, assuming \( X \sim \text{Poisson}(\lambda) \)

Given that \( X \sim \text{Poisson}(\lambda) \), the probability mass function for \( X \) is:
\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

We want the probability that \( X \) is even, i.e., \( P(\text{X is even}) = P(X = 0) + P(X = 2) + P(X = 4) + \dots \)

Define \( P_{\text{even}} \) as the probability that \( X \) is even:
\[
P_{\text{even}} = \sum_{k \text{ even}} \frac{\lambda^k e^{-\lambda}}{k!}
\]

Similarly, the probability that \( X \) is odd, \( P_{\text{odd}} \), is:
\[
P_{\text{odd}} = \sum_{k \text{ odd}} \frac{\lambda^k e^{-\lambda}}{k!}
\]

Now, using the expansions from (b):
\[
e^{\lambda} = \sum_{k=0}^{\infty} \frac{\lambda^k}{k!} = \sum_{k \text{ even}} \frac{\lambda^k}{k!} + \sum_{k \text{ odd}} \frac{\lambda^k}{k!}
\]
\[
e^{-\lambda} = \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{k!} = \sum_{k \text{ even}} \frac{\lambda^k}{k!} - \sum_{k \text{ odd}} \frac{\lambda^k}{k!}
\]

Adding these two equations:
\[
e^{\lambda} + e^{-\lambda} = 2 \sum_{k \text{ even}} \frac{\lambda^k}{k!}
\]
Thus:
\[
\sum_{k \text{ even}} \frac{\lambda^k}{k!} = \frac{e^{\lambda} + e^{-\lambda}}{2}
\]

Therefore, the probability that \( X \) is even is:
\[
P(\text{X is even}) = \frac{e^{\lambda} + e^{-\lambda}}{2 e^{\lambda}} = \frac{1 + e^{-2\lambda}}{2}
\]

---

Would you like further details or explanations? 

Here are five related questions:
1. How is the Taylor series expansion for cosine derived from the exponential series?
2. Can we derive a similar probability for \( X \) being odd if \( X \sim \text{Poisson}(\lambda) \)?
3. What applications might the series \( e^{\lambda} + e^{-\lambda} \) have outside of probability?
4. How would we find a Taylor series expansion for other functions like \( \sin(x) \)?
5. What would change in the calculation if \( X \) followed a different distribution?

**Tip:** Taylor series expansions are powerful tools in approximations, especially for functions that are hard to evaluate directly.

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

Explore the Taylor expansions of exponential functions and determine the probability that a Poisson-distributed random variable is even. This detailed guide covers essential concepts like Taylor Series, exponential properties, and Poisson probabilities, suitable for undergraduate math students.

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