Counterexample bank

False — Two fair coin tosses: A = first is heads, B = second is heads, C = the two agree

False — The standard Cauchy distribution

False — $\max(X_{1}, X_{2})$ with $X_{i}$ ~ Exp(1)

False — X ~ $N(0,1), \varepsilon = \pm1$ with probability ½ independent of $X, Y = \varepsilon{}X$

False — The typewriter sequence on [0,1]

False — The two-state chain that swaps deterministically (period 2)

False — $X_{(n)} ($or $(X_{(1)}, X_{(n)}))$ for Uniform$(-\theta, \theta)$

False — $\sigma$̂$^{2} = (1/n)\sum(X_{i} - {\bar{X}})^{2}$ for $N(\mu, \sigma^{2})$, or $X_{(n)}$ for Uniform$(0,\theta)$

False — $H_{0}: \mu = 0$ vs $H_{1}: \mu \ne 0$ for $N(\mu, 1)$

False — t = $t_{n-1,0.9}$ under the convention $t_{m,\alpha}$ $= (1-\alpha)-$th quantile

False — $Y_{i} = \beta{}x_{i} + \varepsilon_{i}$ with $Var(\varepsilon_{i}) = \sigma^{2}x_{i}^{2}$

False — X ~ N(0,1) and $Y = \varepsilon{}X$ with $\varepsilon = \pm1$ independent

False — A population with a periodic pattern of period k, sampled every k-th unit