I'm never too sure when you are supposed to use uppercase and lowercase letters. Is my usage correct in this solution, also is my use of 'connecting words/phrases' correct?
Question:
Let X be a random sample of size 1 from the Exponential($\lambda$) distribution, where $\lambda \gt 0$ is an unknown rate parameter. The null hypothesis $H_0 : \lambda = 1/2$ is rejected in favour of the simple alternative $H_1 : \lambda = 1$ if the observed value x satisfies $$\frac{f(x; 1/2)}{f(x; 1)} \le \frac{3}4$$ where f(x; $\lambda$) is the PDF of X. Show that the size of the test is $\alpha = \frac{1}3$ and the power of the test at $\lambda = 1$ is $\frac{5}9$
My Solution:
To show that $\alpha = \frac{1}3 $ and $\gamma = \frac{5}9$ at $\lambda = 1$:
Let X be a random sample of size 1 from the distribution $X~Exp(\lambda)$ where $\lambda \gt 0$ is an unknown rate parameter.
Further let $H_0 : \lambda = 1/2$ and $H_1 : \lambda = 1$ be the null and alternative hypothesis, respectively
The pdf of X is given by $$f(x;\lambda)=\lambda e^{-\lambda x}$$
Let T be the test statistic defined by $$T(X)=\frac{f(X; 1/2)}{f(X; 1)}=\frac{\frac{1}{2}e^{-\frac{1}{2}X}}{e^{-X}}=\frac{1}{2}e^{\frac{1}{2}X}$$
Thus the critical region is given by \begin{align} C&=\{X: T(X) \le \frac{3}{4}\}\\ &= \{X: e^{\frac{1}{2}X} \le \frac{3}{2}\} \\ &= \{X: X \le 2\log\frac{3}{2}\} \end{align}
So the size of the test is given by \begin{align} \alpha &= \Bbb{P}_{H_0}(X \le 2\log\frac{3}{2})\\ &= 1- e^{-\frac{1}{2}(2\log\frac{3}{2})} \\ &= 1- \frac{2}{3} = \frac{1}{3} \tag{as required} \end{align}
And the power of the test at $\lambda = 1$ is given by \begin{align} \gamma &= \Bbb{P}_{H_1}(X \le 2\log\frac{3}{2})\\ &= 1- e^{-(2\log\frac{3}{2})} \\ &= 1- \frac{4}{9} = \frac{5}{9} \tag{as required} \end{align}
Upper case typically refers to a random variable, and lower case refers to its realization or grounding, or instantiation.
A statistic is a function of a random variable. Therefore, it is also a random variable. $T(X)$ is a random variable, which is a function of $X$.
Some authors may write the test statistic with its arguments in lower case, but this is assuming that you are already computing a function of the realization of the random variables.
Now, for the critical region, I would assume that it is better to consider lower case variables. The critical region is a deterministic set.
A critical region, also known as the rejection region, is a set of values for the test statistic for which the null hypothesis is rejected. i.e. if the observed test statistic is in the critical region then we reject the null hypothesis and accept the alternative hypothesis.
https://www.ncl.ac.uk/webtemplate/ask-assets/external/maths-resources/statistics/hypothesis-testing/critical-region-and-confidence-interval.html#:~:text=A%20critical%20region%2C%20also%20known,and%20accept%20the%20alternative%20hypothesis.