I'm studying confidence intervals and then I came across the following problem:

It's said that a random variable X has Skewed Exponencial Distribution with parameters $\alpha >0$ and $v \in \mathbb{R}$ when $\alpha (X - v)\text{~}Exp(1)$. Suppose $X_1,X_2,...,X_{20}, Y_1,Y_2,...,Y_{20}$ iid following a Skewed Exponencial distribution with parameters $\alpha > 0\text{ and } v$, both unknown.

a) Let $W_k:=Y_k - X_k, k=1,2,...,20$. Find $E(W_1), Var(W_1) \text{ and } E(W_1^2)$.

b) Let $$Q := \alpha|W_1| + ... + \alpha|W_{20}|$$ Prove that Q is a pivot.

c) Find the Bilateral Confidence Interval for $\alpha$, with covering probability equal 95% based on Q.

I could solve A and B, but I have no idea how to handle c. Does anybody may help me?


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