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I am doing a self-study on Exponential Distribution and noted an exponential on my text giving that:

CDF of Exponential Distribution $$ F(x) = 1 - e^{-λx} , $$ PDF of Exponential Distribution $$ f(x) = λe^{-(λx)} . $$ However, I was wondering on what conditions do I use what?

From what I understand, if I was trying to find the time between consecutive events within a certain period of time, I may use the CDF. However, I am unable about PDF.

Appreciate any advice please.

user1275515
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1 Answers1

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  1. For every real-valued random variable $X$, one can define the CDF of $X$ as the function $$x\mapsto F_X(x)=P(X\leq x)$$ for all $x\in\mathbb{R}$.
  2. Some real-valued random variables, such those with an exponential distribution, are absolutely continuous. This means that there exists a nonnegative function $f$ with the property that $$F_X(x) = \int_{-\infty}^x f(t)\,\mathrm{d}t$$ for all $x\in\mathbb{R}$. This function $f$ is then called the probability density function (PDF) of $X$.
  3. You see that going from the PDF to the CDF requires integration, going the other way differentiation.
  4. To find for example $P(X\in[a,b]$), you can either integrate the density over $[a,b]$ or observe that $P(X\in[a,b])=P(X\in(-\infty,b])-P(X\in(-\infty,a))=F_X(b)-F_X(a)$. This is true because $P(X=a)=0$ in case $X$ is absolutely continuous.

(For completeness, note that the CDF and the PDF of an exponentially distributed random variable are defined to be zero for negative values of $x$.)

binkyhorse
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