Why is the area under the probability density function(PDF) curve gives probability?
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That's pretty much the definition of a probability density function to me. Maybe, could you explain how you think about pdfs? – Callus - Reinstate Monica Sep 27 '13 at 14:41
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The cumulative distribution function (CDF) is defined as a probability, that the random variable $X$ involved is lower or equal than a certain value, $x$, $F_X(x) = P(X\le x)$. This can be extended to $P(x_1\le X \le x_2) = F_X(x_2) - F_X(x_1)$. The probability density function, $f_X(x)$, when it exists, is defined as the derivative of the CDF, $$ \frac {dF_X(x)}{dx}=f_X(x)$$. Then
$$\int_{-\infty}^{x} f_X(x)dx = F_X(x) = P(X\le x) $$
and
$$\int_{x_1}^{x_2} f_X(x)dx = F_X(x_2) - F_X(x_1) = P(x_1\le X \le x_2) $$
So this integral measures probability, and it can be viewed also as measuring "the area under the PDF curve".
Alecos Papadopoulos
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