Hoping you can help me understand the probability density function for the exponential distribution.

Given that the distribution's PDF is described as follows when x > 0:

$$\lambda e^{-\lambda x}$$

Why do we multiply by λ? Sure I'm missing something basic here, but seems that if we're modeling time to event, using e and the exponents would be enough. Is it because we might expect the event to happen multiple times in the time interval?

Thank you in advance.


1 Answer 1


So that when you integrate the pdf between 0 and infinity the probability density sums to 1 as required under the axioms of probability.

  • $\begingroup$ Thank you -- definitely understand that this works, but feel I'm still missing a basic understanding of why this particular way of manipulating the formula makes sense. Have followed proofs through the CDF to the PDF and still feel I'm missing that intuitive understanding. $\endgroup$
    – T_T
    Commented Apr 21, 2014 at 11:59
  • 6
    $\begingroup$ A PDF by definition has to satisfy that its integral over all possible x is 1. The intuition is that of all possible events, something has to happen. Now, when you integrate e^-lambda (without the lambda multiplied beforehand) over 0 to infinity (which are all possible outcomes), you will easily find that it evaluates to 1/lambda. If you then multiply by lambda, you get the desired area (=probability) of 1. $\endgroup$ Commented Apr 21, 2014 at 12:05
  • $\begingroup$ Yes Julian is correct. You also may want to look at this question if you want a deeper understanding of where this distribution comes from: stats.stackexchange.com/questions/2092/… :) $\endgroup$
    – user135784
    Commented Apr 21, 2014 at 12:12
  • 1
    $\begingroup$ Aha, got it! Was missing the integral = 1/lambda. Thank you both. $\endgroup$
    – T_T
    Commented Apr 21, 2014 at 13:31

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