Exponential distribution question

[oneamp]

Hi. I notice that some values of X on the exponential distribution PDF have a value of around 1. I understand the integral ends up being one, since those values of X are less than 1. But P(X) at those points still gets to 1, or thereabouts. How does that make sense, that the probability of a value, say 0.00001, is about 1, and the others complete the integral to 1?

I hope this question makes sense.

Thank you.

 

[FactChecker]

the probability of a value, say 0.00001, is about 1,.

Do you mean the probability of 0<X< 0.00001 is almost 1 or that the PDF at x=0.0001 is about 1?

In the first case, the PDF values must be very large for the integral on the interval 0<X<0.00001 to be nearly 1. Then, yes, there is very little probability of higher values of X.

In the second case the integral over 0<X<0.00001 is about 0.00001. That leaves a lot of probability (0.9999) for higher values of X.

 

[oneamp]

Here's an example of what I'm talking about, from http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm.

The probability in the top left graph actually goes above 1 for some values of X < 1. How is this even possible?

Thank you

 

[homeomorphic]

Imagine a rectangle that is very tall. Say it's 10, 000 meters tall. Does that mean it has a big area?

Well, what if its width is only 0.0000000000000000000000000000000000000000000000001 meters?

Obviously, its area is extremely small, even though it's very tall. If you multiply that tiny number by 10, 000 to get the area, it's still ridiculously small. For essentially the same reason, it's quite possible for a graph to be much taller than 1 for a while, while still having a total area of 1.

 

[oneamp]

Thank you

 

[FactChecker]

Here's an example of what I'm talking about, from http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm.

The probability in the top left graph actually goes above 1 for some values of X < 1. How is this even possible?

Thank you

Remember that those are probability density graphs, not probability graphs. t
The densities are per unit of X. so they can get very large for a short interval of X values. The probabilities themselves never get over 1.

 

[oneamp]

Thank you