Use MLE to calculate exponential distribution parameter

The question is from A Guide to Econometrics by Peter Kennedy (5th edition, page 504.)

Suppose you have a random sample of workers, from several localities,
who have recently suffered, or are suffering, unemployment. For those
currently employed, the unemployment duration is recorded as $x_i$.
For those still unemployed, the duration is recorded as $y_i$, the
duration to date. Assume that unemployment duration w is distributed
exponentially with pdf $lambda e^{-lambda w}$. Find the MLE of
$lambda$.

The official solution is as follows:

I am not sure about the solution, specifically, the PDF of $y_i$. It seems for those who are still unemployed, the probability of observing them unemployed for $y_i$ length is:
$P(Y = y_i) = int_{y_i}^infty P(w)P(y|w)dw$
where $P(w) = lambda e^{-lambda w}$, and $P(y|w)=1/w$, because the events are evenly distributed. In other words, the PDF of $y_i$ is the probability of the entire length being $w_i$, multiplied by the probability of observing a length of $y_i$ conditional on $w_i$. The integral is from $y_i$ to $infty$ because observing a length of $y_i$ means the total duration should at least $y_i$.

Any thoughts on which answer is correct? Thank you!