What exactly is Density and what am I calculating?

[semidevil]

What exactly is "Density" and what am I calculating?

So I"m doing a lot of probability problems with CDF and although I know how to calculate it, I don't quite know what I am calculating (in terms of relating it to real life situations).

Example:
lets say the lifetime of a machine part has a continuous distribution of (0,40) with a density function that is proportional to (10x+2)^-2. What is the probability that the lifetime of a part is less than 5.

So I know how to solve this: integrate, find the density, and we get a value for C. Then, we integrate again to between 0 and 5 to get the probability that it is less then 5.

First of all, what is (0, 40) when they say there is a distribution between that? what does it mean that the probability of less than 5 is 5/12 (the correct answer)?

since 5 is on the lower end of (0, 40), is it saying that the part has a very low lifetime?

 

[Simon Bridge]

Example:
lets say the lifetime of a machine part has a continuous distribution of (0,40) with a density function that is proportional to (10x+2)^-2. What is the probability that the lifetime of a part is less than 5.

So I know how to solve this: integrate, find the density, and we get a value for C. Then, we integrate again to between 0 and 5 to get the probability that it is less then 5.

First of all, what is (0, 40) when they say there is a distribution between that?

It means that P(0<x<40)=1
For your example, the part will certainly fail before it turns 40, but after it leaves the production line (at t=0).

what does it mean that the probability of less than 5 is 5/12 (the correct answer)?

That means that there is a 5/12 probability that the part won't turn 5.
You would expect, in a big production run, that 5 out of every 12 parts will have failed by 5 time-units after they were made.

since 5 is on the lower end of (0, 40), is it saying that the part has a very low lifetime?

That depends - if the required lifetime is only 1 time-unit, this is a long time.
It just means you'd expect almost half of them not to be really long lived.

Think cars - with careful maintenance a car can last for a century or more, but most get trashed before they are a decade old and you are lucky to get a guarantee longer than 18 months. Does that mean cars don't last very long?

 

[Stephen Tashi]

So I"m doing a lot of probability problems with CDF

You mention "density" in the title of your post. Do your materials call the CDF a "cumulative density function"? Many texts call the CDF a "cumulative distribution function". It's derivative is the "density".