2
votes
Commutative diagram involving order statistics
I don't think there is anything special about random variables or CDFs here. For any list of distinct [non-random] numbers $x_1, \ldots, x_n$ and any increasing function $f$, we have $\text{sort} \...
- 91.2k
2
votes
Accepted
Proving that Kurtosis is bounded from below by skewness plus $1$
Let $ Z := (X-μ)/σ$ with $μ := 𝔼[X]$ and $σ := \sqrt{\mathrm{Var}{(X)}}$. We want to establish the inequality $$ 𝔼[Z^4] \geq 1 + 𝔼[Z^3]^2.$$
Note that $𝔼[Z] = 0$ and $𝔼[Z^2] = 1$. Hence $𝔼[Z^3] ...
- 750
1
vote
Commutative diagram involving order statistics
It works because the cdf $F$ is an increasing function, that is:
$$X_1 < X_2 \iff F(X_1) < F(X_2)$$
- 16.8k
1
vote
Accepted
Proving Negative Covariance Between a Component of a Random Vector and the Reciprocal of Its Sum
Suppose $X=(X_1,X_2)$ is $(1,7)$ with probability $\frac12$ and is $(3,1)$ with probability $\frac12$.
Then $\frac1Z=\frac{1}{X_1+X_2}$ is $\frac18$ in the first case and $\frac14$ in the second case
...
- 159k
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