Assume that the salary of a randomly chosen individual has the exponential distribution with unknown parameter $λ$. Denote by $X_1, . . . , X_n$ the binary responses about salary being bigger than some threshold $(X_i ∈ {0, 1}, i = 1, . . . , n)$ of the $n$ sampled individuals.
Is it $Bern(1-e^{λy})$?
EDIT: Sorry, Let $Y_1, . . . , Y_n$ be the salaries of the $n$ sampled people.
For $X_i = 1$ we have that:
$$P(X_i= 1) = P(Y_i > T) = \int_T^{+\infty}\lambda e^{-\lambda y}dy = e^{-\lambda T}.$$
Similarly, for $X_i=0$ we have that:
$$P(X_i= 0) = P(Y_i < T) = \int_0^{T}\lambda e^{-\lambda y}dy = 1 - e^{-\lambda T}.$$
Then, $X_i$ is a Bernoulli distribution with parameter $p = e^{-\lambda T}$ ($p$ is the probability that $X_i = 1$).