User:Inyuki/LaTeX notes

D

Question 1

Q: Draw $5$ random numbers from $\chi _{2}^{2}$ distribution, using its definition.

A: By definition, $X\sim \chi _{2}^{2}\Leftrightarrow X=Z^{2}+Z^{2}$, where $Z\sim {\mathcal {N}}(0,1)$.

According to CLT, and some simulation, we see that:

if $U_{i}\sim U(0,1)$ are from uniform distribution,

then approximately $\sum _{i=1}^{12}U_{i}-6{\underset {\mathrm {approx.} }{\sim }}{\mathcal {N}}(0,1)$.

So, we can draw observations from $\chi _{2}^{2}$ by $Z^{2}+Z^{2}\approx \left(\sum _{i=1}^{12}U_{i}-6\right)^{2}+\left(\sum _{i=1}^{12}U_{i}-6\right)^{2}$

Question 2

What is Simpson's paradox?

P-value

The thing is, that

${\textrm {p{\mbox{-}}value}}\Leftrightarrow P(T\leq t|H_{0})$ or $P(T\geq t|H_{0})$

is not equivalent to:

${\textrm {p{\mbox{-}}value}}\Leftrightarrow P(T\in K|H_{0})$

Where $T$ is the test statistic, and $K$ is critical region, and $t$ is the obtained value of test statistic.

I guess, the right definition is in german wikipedia:

In case of right-sided test:

$p_{\text{right}}=P(T\geq t|H_{0}).$

In left-sided test:

$p_{\text{left}}=P(T\leq t|H_{0}).$

In case of two-sided test:

$p=2\cdot \min(p_{\text{right}},p_{\text{left}}).$