In probability theory and statistics, the law of the unconscious statistician (LOTUS) is a theorem used to calculate the expected value of a function g(X) of a random variable X when one knows the probability distribution of X but one does not know the distribution of g(X). The form of the law can depend on the form in which one states the probability distribution of the random variable X. If it is a discrete distribution and one knows its probability mass function ƒX (but not ƒg(X)), then the expected value of g(X) is
E
[
g
(
X...
user12692
user12692
All you need to know is that $3 < \pi < 4$ and $e > 2$.
As $n \ge 1$ you have $1 + \dfrac 1{2n} \le \dfrac 32$ so that $\dfrac 2\pi \left( 1 + \dfrac 1{2n} \le \dfrac 32 \right) \le \dfrac 3 \pi < 1$,
and $\ln \pi < \log_2 4 = 2$.
user12692
user12692
Claim If $L_n=\displaystyle\frac{1}{\pi}\int_0^{\pi}|D_n(t)|dt$, we have $L_n=\dfrac{4}{\pi^2}\log n+O(1)$.
Proof We begin by dividing the interval $[0,\pi]$ into the subintervals where $\sin\left(n+\dfrac 1 2\right)$ keeps its sign. Since $\sin t/2$ is always non-negative in said domain, we...
Suppose $Γ$ derives $A→¬A$ and $¬A→A$ then $Γ$ derives $A$ and $¬A$.
Also how to prove : $∀α, α ∈ ∆$ iff $¬α∉ ∆$ where $∆$ is maximal consistent set.
