Suppose we have two independent random variables $X_1 \sim \exp(\lambda_1)$ and $X_2 \sim \exp(\lambda_2)$ .
Now, we are given just one observation each from the two distributions above, say $S_1$ and $S_2$ respectively. On the basis of these two observed numbers $S_1$ and $S_2$ , I am interested in testing the following two different hypotheses, viz.
- Null hypothesis $~~H_0 : ~~\lambda_1 = \lambda_2~~$ against the alternative hypothesis $~~H_A : ~~\lambda_1 \neq \lambda_2$
- Null hypothesis $~~H_0 : ~~\lambda_1 = \lambda_2~~$ against the alternative hypothesis $~~H_A : ~~\lambda_1 < \lambda_2$
Can someone please suggest appropriate ways to test these two hypotheses at some given level, say $\alpha\in(0,1)$ ?
Note : I guess, under the assumption of the null hypothesis (in both the problems), we may play with the random variable $Z = \frac{X_1}{X_1+X_2}$ because it'll probably be uniformly distributed. Anyway, I'm unable to think any further. Also, here $\exp(\lambda)$ refers to an exponential distribution with parameter $\lambda$ .