I am currently trying to learn about the exponential family of distributions. I am trying to understand this question and this answer from Xi'an. I have the same function:
$$f(x; \sigma, \tau)= \begin{cases} \dfrac{1}{\sigma} e^{-(x - \tau)}/\sigma &\text{if}\, x\geq \tau\\ 0 &\text{otherwise} \end{cases}$$
However, it still isn't clear to me why this is not a member of the exponential family. On the other hand, it is said that the function
$$f_{\varphi}(x)= \begin{cases} \varphi x^{\varphi - 1} &\text{if}\, 0 < x < 1, \varphi > 0\\ 0 &\text{otherwise} \end{cases}$$
actually is a member of the (one-parameter) exponential family.
Why is the first function not a member of the exponential family? Furthermore, what is the reasoning behind why one is a member of the family and yet the other is not?
The 'duplicate' does not provide the clarification of the reasoning here Is the negative exponential distribution a member of the exponential family? and here Is the negative exponential distribution a member of the exponential family?. I was really looking for a 'simpler' explanation of things so that I could understand.
