Consider the stochastic process $\{X_t\}$ such that $\mathbb{P}(X_0=1)=\mathbb{P}(X_0=-1)=1/2$. $X_t$ changes sign in Poisson times.

Question

Consider the stochastic process $\{X_t\}$ such that $\mathbb{P}(X_0=1)=\mathbb{P}(X_0=-1)=1/2$. $X_t$ changes sign in Poisson times, that is, the probability of $k$ changes of sign in a time interval of length $T$ is $\frac{(\lambda T)^{k}e^{-\lambda T}}{k!}$ where $k$ is even. What is the probability that there are $k$ changes of sign in an interval of length T?

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