To show that $P(T_0 = k) = P(T_1 = k - 1)$ for $k = 2, 3, \ldots$, we can use an appropriate application of the law of total probability involving $X_1$, the first step of the process.

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Explanation:

1. Define the random variables:

   • $T_0$: The first time the process reaches height $0$.
   • $T_1$: The first time the process reaches height $1$.
   • $X_1$: The value of the process after the first step.

   The law of total probability allows us to break down the probabilities of $T_0$ and $T_1$ by conditioning on the first step $X_1$.

2. Key property of the process: Suppose the process starts at height $1$. Then:

   • To reach height $0$ for the first time at step $k$, the process must reach height $1$ for the first time at step $k - 1$.

   Mathematically, this means: $\{T_0 = k\} = \{T_1 = k - 1\}$

3. Transitioning from height 1: When the process starts at height $1$, we consider the contribution of all possible outcomes of $X_1$ (the value of the process after the first step).

   By conditioning on $X_1$, the law of total probability gives: $P(T_0 = k) = \sum_{x} P(T_0 = k \mid X_1 = x) \cdot P(X_1 = x)$

4. Connection to $T_1$: Similarly, the probability $P(T_1 = k - 1)$ can be written as: $P(T_1 = k - 1) = \sum_{x} P(T_1 = k - 1 \mid X_1 = x) \cdot P(X_1 = x)$

   Since $\{T_0 = k\}$ and $\{T_1 = k - 1\}$ describe the same event under the key property above, their probabilities are equal: $P(T_0 = k) = P(T_1 = k - 1)$

5. Range of $k$: This relationship holds for $k = 2, 3, \ldots$ because for $k = 1$, $T_0$ does not align with $T_1 + 1$.

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Final Justification:

The equality $P(T_0 = k) = P(T_1 = k - 1)$ follows directly from the recursive structure of the process and the law of total probability applied to the outcomes of $X_1$, ensuring the first event of height $1$ corresponds to the first event of height $0$ shifted by one step.

Would you like more detailed steps, or an example to illustrate this?

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Follow-Up Questions:

1. Would you like an example of a stochastic process (e.g., a random walk) to solidify this concept?
2. Do you want the derivation broken down further in terms of conditional probabilities?
3. Would you like an explanation of the significance of the range $k = 2, 3, \ldots$?
4. Should I illustrate how the law of total probability applies specifically to $T_0$ and $T_1$?
5. Do you want to extend this result to other height levels or general stopping times?

Tip: Always start by identifying the key recursive properties of the random process—this makes it easier to apply probabilistic tools effectively!