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 Nov26 comment Finding the density of rayleigh distribution $T_4$ doesn't mean $x=4$! $T_1,T_2,..T_{20}$ are 20 independent random variables each following a Rayleigh distribution: $f_{X}(x) = xe^{-x^2/2}$ and $f_{Y}(y) = ye^{-y^2/2}$. Nov25 comment Finding the density of rayleigh distribution The first one comes from this: $\text{min}(T_1, T_2, T_3, ... T_{20}) \lt t$ is the "opposite" of all 20 $T_i$'s being greater than $t$. The second comes from X and Y being independent; being independent implies $P(X\dfrac{e^x}{2}$ @math110, it is the same. You only need to show $n! > 2e^{-\lambda n} \int^{\lambda n}_0 (\lambda n -t)^n e^t dt$ which follows basically what you have already. May4 comment prove this inequality$1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^n}{n!}>\dfrac{e^x}{2}$ @math110, I meant replace your argument of $x=n$ with $x=\lambda n$ and you will arrive at the same conclusion you have for $x=n$. May4 comment prove this inequality$1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^n}{n!}>\dfrac{e^x}{2}$ I think you can do this by letting $x := \lambda n$ where $0 \lt \lambda \le 1$. All your inequalities will remain true. Jan16 comment Lost on algebra notation @Clayton, I think that's an exaggeration. Most undergrad number theory books don't require much algebra prerequisites if any (eg, old books like Landau and Hardy/Wright).