# Random stuff 2026 week 3
Fix a positive real number $N$. Suppose $x,t$ are positive reals such that $xt = N$ with $x^{t}$ maximized. What is $x$?
Since $t = N / x$ we have $x^{N/x} = e^{ \frac{N}{x} \ln(x) }$ is maximized. So $[\frac{-N}{x^{2}}\ln(x) + \frac{N}{x^{2}} ] x^{N/x} = 0$. So $\frac{N}{x^{2}}[1-\ln(x)]=0$, or $\ln(x)=1$. In other words $x=e$.
This gives a characterization of $e$, a positive real $x$ that maximizes $x^{t}$ if $xt = N$ for some fixed positive $N$.
What if $x^{t} = N$ is fixed instead. This time assume $x > 1$ and $xt$ is minimized. What is $x$?
In this case $t \ln(x) = \ln(N)$, so $\frac{x \ln(N)}{\ln(x)}$ is maximized. So $\ln(N) \frac{\ln(x)- 1}{\ln(x) ^{2}} = 0$. This means $\ln(x)=1$, or $x=e$ as well!
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Is random variable a measurable function, or an equivalence class of measurable functions, where two measurable functions are in the same class if they are equal almost everywhere?
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