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I am reading the following paper on page 15 it is written that for Weibull distribution with scale and shape parameters estimated at 13.6 and 2.6, they estimate the following probabilities: $$p_1=P(W<7)=0.18$$ $$p_2=P(7<W<14)=0.51$$ $$p_3=1-p_1-p_2=P(14<W<21)=0.31$$ I know that the Weibull distribution is given by: $$f(x)=\frac{k \left(\frac{x}{\lambda }\right)^{k-1} \exp \left(-\left(\frac{x}{\lambda }\right)^k\right)}{\lambda }$$ so for $k=2.6$ and $\lambda=13.6$ we should be able estimate $f(x)$ or the probability. I know that for fixed $x$ this is simple for example $f(10)=0.07$. I am trying to understand the notation $P(W<7)$, $P(7<W<14)$ and $P(14<W<21)$ how does one calculate the probabilities in that case?

Wiliam
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    The function $f$ is a probability density function (PDF), which you can think of as a kind of "continuous histogram" for the distribution. A histogram tells you, for any value the random variable takes on, the probability that it takes on that value. A PDF tells you a variation of that. The probability that a random variable with a Weibull distribution takes on any given value is $0$, so what the PDF tells you is the probability per unit value. I.e., if you were to find the probability that the variable equals somewhere between $9.5$ and $10.5$, it would be very close to $f(10)$. – Brian Tung Jun 06 '20 at 03:20
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    In order for you to find the probability that the random variable falls within a given interval, you need to integrate the PDF between the bounds of that interval. Conveniently, the antiderivative of the PDF (calibrated so that it always falls between $0$ and $1$) is the cumulative distribution function (CDF), typically denoted $F(x)$. Then, for example, $F(14) - F(7) = P(7 \leq X \leq 14)$. – Brian Tung Jun 06 '20 at 03:21

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The probabilities are found by integrating the density function over the interval of interest. For example, $P(W<7) = \int_0^7 f(x)\, dx$ and $P(7<W<14) =\int_7^{14} f(x)\, dx$.

Note that the Weibull CDF is $\int_0^x f(t)\, dt = 1 - e^{-(x/\lambda)^k}$ for $x\ge 0$, so we have $P(a < W < b) = F(b)-F(a)$ for all $0\le a \le b$, with $F(x)= 1 - e^{-(x/\lambda)^k}$.

Minus One-Twelfth
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