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We are tasked with analyzing the process \( X(t) = N(t) - \lambda t \), where \( N(t) \) is a Poisson process with rate \( \lambda \). Let's solve for the required expectations.

### 1. Finding \( \mathbb{E}[X(t)] \)

Since \( N(t) \) is a Poisson process with parameter \( \lambda t \), we know:
\[
\mathbb{E}[N(t)] = \lambda t.
\]
Thus, for \( X(t) = N(t) - \lambda t \):
\[
\mathbb{E}[X(t)] = \mathbb{E}[N(t)] - \lambda t = \lambda t - \lambda t = 0.
\]

### 2. Finding \( \mathbb{E}[X(t) \mid X(s) = x] \), where \( s < t \)

We need the conditional expectation \( \mathbb{E}[X(t) \mid X(s) = x] \). To analyze this, note that:
\[
X(t) = N(t) - \lambda t, \quad X(s) = N(s) - \lambda s.
\]
This implies:
\[
N(t) = X(t) + \lambda t, \quad N(s) = X(s) + \lambda s.
\]
Since \( N(t) \) is a Poisson process, the increments \( N(t) - N(s) \) are independent of \( N(s) \) and follow a Poisson distribution with parameter \( \lambda (t-s) \). Therefore:
\[
N(t) - N(s) \sim \text{Poisson}(\lambda (t-s)).
\]
Using this, we write:
\[
X(t) = N(t) - \lambda t = \big(N(s) + (N(t) - N(s))\big) - \lambda t.
\]
Substituting \( N(s) = X(s) + \lambda s \), we get:
\[
X(t) = (X(s) + \lambda s) + (N(t) - N(s)) - \lambda t.
\]
Simplify:
\[
X(t) = X(s) + (N(t) - N(s)) - \lambda (t-s).
\]
Now take the expectation conditional on \( X(s) = x \). Since \( N(t) - N(s) \) is independent of \( N(s) \), and its mean is \( \lambda (t-s) \):
\[
\mathbb{E}[X(t) \mid X(s) = x] = x + \mathbb{E}[N(t) - N(s)] - \lambda (t-s).
\]
Substitute \( \mathbb{E}[N(t) - N(s)] = \lambda (t-s) \):
\[
\mathbb{E}[X(t) \mid X(s) = x] = x + \lambda (t-s) - \lambda (t-s) = x.
\]

### Final Results
1. \( \mathbb{E}[X(t)] = 0 \).
2. \( \mathbb{E}[X(t) \mid X(s) = x] = x \).

Let me know if you want further clarifications!

### Relative Questions
1. How does the independence of Poisson increments influence the conditional expectation?
2. What would change if the Poisson process had a time-varying rate \( \lambda(t) \)?
3. Can we generalize this result for a different transformation of \( N(t) \)?
4. How does this relate to martingale properties of \( X(t) \)?
5. What are practical scenarios where such processes might be modeled?

### Tip
Always consider the independence and distributional properties of increments when analyzing Poisson processes.

Let N(t) be a Poisson process with parameter λ. Define a new process X(t) = N(t) - λt. Find E[X(t)] and E[X(t) | X(s) = x] where s < t.

Learn how to compute the expectation and conditional expectation for a transformed Poisson process X(t) = N(t) - λt. This detailed solution explores key concepts such as the independence of Poisson increments and conditional expectation. Suitable for advanced probability students.

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