Exponential distribution question

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Discussion Overview

The discussion revolves around the properties of the exponential distribution, particularly focusing on the interpretation of the probability density function (PDF) and how it relates to probabilities for specific values of X. Participants explore the implications of PDF values exceeding 1 and the relationship between the PDF and the integral that sums to 1.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how the PDF can yield values around 1 for certain X while the integral of the PDF equals 1, expressing confusion about the probability of very small values of X being close to 1.
  • Another participant seeks clarification on whether the original question pertains to the probability of a range (0
  • It is suggested that if the PDF values are large for small intervals, the integral over that interval could approach 1, leaving significant probability for larger values of X.
  • A participant provides an analogy involving a very tall rectangle with a very small width to illustrate that a high PDF value does not necessarily imply a large area under the curve.
  • Another participant emphasizes the distinction between probability density graphs and probability graphs, noting that while densities can exceed 1, the actual probabilities do not.

Areas of Agreement / Disagreement

Participants express differing interpretations of the PDF and its implications, particularly regarding values exceeding 1. There is no consensus on the initial question about the probability of specific values of X, and the discussion remains unresolved.

Contextual Notes

Participants highlight the importance of understanding the distinction between probability density and actual probability, which may depend on the definitions and interpretations used in the discussion.

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

rhpis6.jpg


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

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

rhpis6.jpg


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.
 
Thank you
 

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