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Let $\delta >0$ be the delay in your departure time from work. Also, we can model your remaining commute time $T$ as a function of the current time and your current position $(t,x_t): T(t,x_t)$. We will assume that $x_t$ is continuous (no teleportation allowed!) and monotonically increasing (no backtracking). Your coworkers' conjecture (let's call it ...


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The notation $\Sigma$ (read "sigma") is a capital "S" from the greek alphabet. It is generally used to denote a sum. Indeed, it is interesting to have shorthand for very long sums or to highlight some quantities. If you have $$\sum_{i=1}^{n}$$ The notation $i=1$ means that we count with respect to the indice $i$ and that this indice begins from $1$. The ...


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Maybe this is a statistics exercise. You measure the hight and femur length in a number of males, then plot the data in 2D. Perhaps you can fit a straight line to the plot.


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HINT This website makes a claim that In an adult, the femur is said to be approximately a quarter of a person’s height. So given the length $x$ of the femur, how would you find the height of the entire person?


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Don't think this has any solutions as a functional equation. From $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)} $, if $m=t+1$, this gives $\frac{f(s, m+1)}{f(s, m)} = \frac{1}{1+s} $. Applying this twice, $\frac{f(s, m+2)}{f(s, m)} = \frac{1}{(1+s)^2} $. However, setting $m = t+2$ gives $\frac{f(s, m+2)}{f(s, m)} = \frac{1}{1+2s} $. This is a ...


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A fixed-effects model may be useful here. Let $t_{pcrn}$ be completion time for a given person $p$ driving in car $c$ on track $r$ for run $n$ (assuming there can be multiples of each triplet $pcr$. We could model $t_{pcrn}$ using a three-factor, additive, fixed-effects model: $$t_{pcrn} = \mu + \phi_p+\alpha_c + \beta_r + \epsilon_{pcrn}$$ Where: $\mu$ ...


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Perhaps you could try something like $$f(x) = - (1 + e^{-cx}) \arctan \left( \frac{\sin(x)}{1 + e^{-cx} - \cos(x)} \right) $$ where $c > 0$.


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Assuming both Poisson processes are independent and all buses have infinite capacity which is able to let all people waiting at the stop to enter the bus. Let $T_k$ be the arrival time of the $k$-th bus and $X(t)$ be the cumulative number of arrivals of the people at the stop. Then $T_k \sim \text{Gamma}(k, \mu)$ and $X(t) \sim \text{Poisson}(\lambda t)$, ...


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Hint: I would prefer the following differential equation: $$\frac{dI(t)}{dt}=k\cdot (1,000,000-I(t))-600$$ The more people have been infected the less new infections can happen.



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