Interview problems: counter examples in probability

Problems:
(1) Zero correlation does not imply independence.

(2) For two marginally normal random variables, zero correlation does not imply independence.

(3) Pairwise independence does not imply jointly independence.

(4) For three events A, B, C: A is independent with B and A is independent with C. Is A independent with the intersection of B and C?

(5) Convergence in distribution does not imply convergence in probability.

(6) Convergence in probability does not imply almost sure convergence.





Solution:
(1) X: take values of -1, 0, +1 with equal probability. $Y=X^2$.

(2) X is standard Gaussian. Let W = +1 or -1 with equal probability and W is independent with X. Define Y=WX. X and Y are standard Gaussians. They are uncorrelated but dependent.
Note: Only for random variables with jointly Gaussian, zero correlation implies independence.

(3) Throw two dices. A: sum is 7, B: get 3, C: get 4.

(4) No. Throw a dice. A: get a even number, B: get 1 or 2, C: get 2 or 3.

(5) Let $\Omega=\{0, 1\}$ be two possible outcomes. $X_n(\omega=0)=X(\omega=1)=0$ and $X_n(\omega=1)=X(\omega=0)=1$. $F_{X_n}(x)=F_{X}(x)$ but $|X_n - X|=1$.

(6) Let the sample space be the uniform $[0, 1]$ and define the sequences: $X_1=I_{[0, 1/2]}$,  $X_2=I_{[1/2, 1]}$, $X_3=I_{[0, 1/4]}$, $X_4=I_{[1/4, 1/2]}$, $X_5=I_{[1/2, 3/4]}$, $X_6=I_{[3/4, 1]}$, $\cdots$. $X_n$ converges to 1 in probability but not almost sure.



Location: 美国加利福尼亚州帕罗奥图

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