Let $θ$ be an unknown constant. Let $W_1,…,W_n$ be independent exponential random variables each with parameter 1. Let $X_i=θ+W_i$.
How do I calculate the distribution of $X_1$, given θ? Notice that I don't look for a conditional probability here since $θ$ is a constant. My guess is that $P_X(x;θ)= e^{-(x_1-θ)}$ but I'm not sure. Thank you.
Best Answer
You are correct. To prove it, just plug it in the definition of cdf:
$$P(X_i \le x) = P(W_i \le x - \theta_i) =( 1 - e^{-(x - \theta_i)})1_{x \ge \theta_i}$$
If you want the pdf you can take the derivative to get
$$f_{X_i}(x) = e^{-(x-\theta_i)}1_{x \ge \theta_i}$$