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1) $Y=F(X)$ is a transformation applied on $X$, just like $Y=X^2$ or $Y=\sqrt{X}$. So, $F(X)$ is not the PDF of $Y$, it's a transformation applied on $X$. The author chooses this transformation function to be the CDF of $X$ (for some reason of course, but that's not the question here). So, that is not the PDF of $Y$ (if it is, where is $y$ in it?). We just ...


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From comment while question was closed: What is possible is a Poisson process with a rate which varies over time. That could give the sort of graph you are seeing on Google. As you commented, this would be a nonhomogeneous Poisson process If you integrate that rate over a particular time period, then the actual number of arrivals in the whole of the time ...


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According to Formula 26.3.19 of Abramowitz and Stegun's Handbook of Mathematical Functions, $$\int_0^\infty\int_0^\infty f(x,y;\rho)\;\mathrm dx\;\mathrm dy = \frac 14 + \frac{\arcsin \rho}{2\pi}$$ where $f(x,y;\rho)$ is the bivariate normal density of two standard normal random variables with correlation coefficient $\rho$. By symmetry, $\frac 14 + \frac{\...


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Take independent standard normal random variables $X$ and $Y$. Then the joint distribution of $X$ and $\rho X-\sqrt{1-\rho^2}\,Y$ is the same as joint distribution of $\epsilon_1$, $\epsilon_2$. This is Cholesky's decomposition. Then $$ \mathbb P(\epsilon_1\leq 0, \epsilon_2\leq 0) = \mathbb P\left(X\leq 0, \rho X-\sqrt{1-\rho^2}\,Y \leq 0\right)=\mathbb P\...


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A common way to measure inequality is the Gini coefficient as one of many metrics. However, depending on your concern and your business problems, you might want to define a metric on your own. For example, if the revenue concentration on too few customers causes problems in the financial balance of the company, one can estimate a turnover probability (...


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Let's write the calculus formally. $$f(x) = \begin{cases} 160x^{-6} & x\ge2\\ 0 & x< 2 \end{cases} $$ To get from $f(x)$ to $F(x)$, integrate: $$F(x) = \int_{-\infty}^{x}f(t)dt.$$ But we can break up the integral into an integral from $-\infty$ to $2$ and another from $2$ to $x$. $$ F(x) = \int \limits_{-\infty}^{x} f(t) \ ...


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