Iām Thao. Iām trying to see the world with my own eyes and think for myself, and this website is where I practice and share what I find meaningful.
Monotonicity
Question: Show that if a relation is complete, transitive and satisfies the independence axiom, then it satisfies the monotonicity property. Answer: Without loss of generality, assume $$ p \succ q $$. We need to show that for any probability a and b, $$ ap + (1-a) q \succ bp + (1-b) q $$ iff $ a >b $ Rewrite the left hand side as $$ (a-b)p + bp + (1-a) q$$....
P-Hacking
P-value is the probability of witnessing an event at least as extreme as our result. The smaller the p-value, the more evidence there is to reject \(H_0\). Reduce p-value if you want to be extra-sure that you have good evidence to reject \(H_0\). p-hacking collect a bunch of data test a bunch of different things and look for p-values <0.05 claim strong evidence for the things that have p-values <0.05 $$ P(\text{at least one of the $S_i$ has a p-value below significance}|H_0) = 1 - P(\text{none}|H_0) = 1- 0....
Sampling variance vs population variance
Population variance measures the variability of the underlying variable $$ V(Y_i) = E(Y_i - EY_i)^2. $$ Sampling variance measures the variability of the sample mean in repeated samples $$ V(\bar{Y}) = E(\bar{Y} - E\bar{Y})^2 = E( \bar{Y} - EY_i)^2 = \dfrac{\sigma_Y^2}{n}.$$ Sampling variance depends on sample size. More data means less dispersion of sample averages in repeated samples (Law of Large Number). We often work with standard error of the sample mean which summarizes the variability in an estimate due to random sampling $$ SE(\bar{Y}) = \dfrac{\sigma_Y}{\sqrt{n}} $$...
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