$X$ is a random variable and its PDF is:
$$ f(x;θ) = \begin{cases} θe^{-θx}, & x > 0 \\\ 0, & x \le 0 \end{cases} $$
With a constant $θ_0 > 0$:
$$ H_0: θ = θ_0 \\\ H_1: θ < θ_0 $$
with significant level $\alpha \quad (0 < \alpha < 1)$.
Now, with a constant $d$, $Y$ is a random variable based on $X > d$.
That is to say, for every $y > d, P(Y > y) = P(X > y | X > d)$. Set $b' > d$, when $Y > b'$, we will reject $H_0$.
To find the constant $b'$ which is the reject region value in this case, and based on the memoryless of the exponential distribution, I tried to write:
$$ P(Y > y) = P(X > y | X > d) = P(X > y - d + d| X > d) = P(X > y - d) $$
Typically, with left-side testing, I would write:
$$ \dfrac{b' - θ}{\sqrt{1/θ^2n}} = -ϕ^{-1}(\alpha) $$
Yet, I am not sure how to do such a similar step on a conditional probability.