I'm confused as to how I should be interpreting the exponential distribution and its probability density function (PDF). I'm specifically referring to the fact that it peaks close to $x = 0$ and then tapers off.
Does the PDF of an exponential distribution indicate that (some type of) machinery is MOST likely to fail immediately after it is built? In other words, does it indicate that the probability of failure is highest right after the machinery has been built? This doesn't seem to intuitively be correct, since machinery should have the least chance of failure if it is brand new, right? This makes me suspect that I'm misunderstand something.
Example #1:
Example #2:
Example #3:
Take examples #2 and #3. Example #2 has had the PDF superimposed on top of it. Since the PDF is largest close to $x = 0$, how do I interpret this? If I compare it to example #3, then it would be saying that the probability is highest that the time to failure is small -- somewhere between 0-80? But that would mean that the machinery is most likely to fail soon after it is brand new! And so it doesn't seem to make intuitive sense.
I'm very confused as to how I should be interpreting this, and I would greatly appreciate it if people could please take the time to clarify.
The second graph is wrong in that it shows a vertical tangent at $\text{time}=0.$
An exponential distribution is the right model only if the conditional probability of failure on the $n\text{th}$ day given that it has survived until then, is the same as the probability of failure on the first day.
However, that implies that failure on the $n\text{th}$ day is less probable than failure on the first day (if $n>1$) only because of the possibility that the machine already failed before the $n$th day.