Exponential Distribution - Memoryless and examples In several (introductory) statistics books we can see that they use an Exponential Distriubtion to model the time of failure of an electronic component. I understand that it got the appealing property of being bounded $[0 , \inf]$ , however I have a concern because of the memoryless property of this distribution.
I have the following questions:
1) If I have the assumption that an electronic component that I already used a lot of time will have less chance to "survive" than one that I used only a few hour. In other words $P(T > a + b | T > a) < P(T > b)$ I'm breaking the memoryless assumption , then the Exponential Distribution is not a good choice. Am I right ?
2) What distribution would you suggest to take that into account? I'm thinking about a Truncated Normal $[0,\inf]$, however it's possible that due to factory problem we have many products that won't work at all, so maybe it's not the ideal choice for this use case.
3) Maybe electronic components have a certain behavior that makes the memoryless property a realistic assumption ?  but I really would like to have answer for the question 2) by maybe changing the problem to the number of km before having a motor issue in a car or any example where the wear impacts negatively the time to survive.
 A: 1) Your lay description isn't precise enough to say. We can say all components will fail eventually. So the chance of surviving is 0 regardless of how long the product already been in use. In your notation however, you correct note a process with some kind of memory. Here you are interested in the $b$-hour survival. Indeed, the exponential distribution will not describe well a process with the probability rule you note. 
2) The Weibull distribution is a generalization of the exponential model with a shape and scale parameter. The hazard is linear in time instead of constant like with the Exponential distribution.
3) Collect data, conduct a 1-degree of freedom likelihood ratio test for the Weibull vs Exponential model. If the scale parameter is statistically significant, conclude that the distribution is not memoryless.
A: *

*You're right, exponential distribution is not a good choice in that case. It doesn't model aging properly.

*Weibull distribution might be appropriate. Read the case $k>1$ in the wiki link. For the products that won't work at all, you can zero inflate any distribution you have by mixing with a Bernoulli.

*I haven't encountered a case specific to a single electronic component that using memoryless property is valid. But, sometimes a population of components can be modelled quite well with simpler assumptions like this one.

