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

Comments

Post a Comment

Popular posts from this blog

529 Plan

This post is for knowledge sharing only. It is not intended to be investment or tax advice. 529 Plans are tax-advantaged accounts specific for education saving: Contribution are made with after-tax dollars. Growth within the account is tax-free. Withdraw for qualified education expenses is also tax free. Roth IRA First 529 Plans are very similar to Roth IRA . But Roth IRA withdraw under  the 5-year rule  is tax free for any purpose. Therefore, unless for the state tax deduction in some states, it is always better to max out Roth IRA first (through backdoor and mega backdoor  for high income earners) before funding a 529 plan. Note: there is NO California state tax deduction for any 529 plan. 529 or Not? Here are some considerations in reality before funding a 529 plan: What if no qualified education expenses In the worst case that there will be no qualified education expenses, you can still take money out for non-qualified expenses, but you have to pay: federal income tax + 10% federa
Read more

How to offset W2 tax

This post is for knowledge sharing only. It is not intended to be tax or investment advice. 1. Contribute to pre-tax 401K and HSA As well known, pre-tax 401K and health saving account (HSA) are the only a few tax shelters for W2 workers.  2. Tax-loss harvesting If your total capital losses exceed your capital gains in a tax year, you can deduct up to \$3,000 (married filing jointly) from your ordinary income. If your losses are more than $3,000, you can carry over the remaining amount to future years, continuing to offset your income until all the losses are used up. A popular method to generate these deductible losses is  tax-loss harvesting . This involves selling a security at a loss to offset gains or reduce ordinary income, and optionally purchasing a similar, but not "substantially identical", security to avoid missing out on the market. However, be aware of the  wash-sale rule , which disallows claiming a loss if you buy the same or a substantially identical security
Read more

Health Saving Account (HSA)

This post is for knowledge sharing only. It is not intended to be investment or tax advice. Health saving account (HSA) is another tax-free investment account that is even better than Roth IRA . HSA offers three tax advantages: The contribution is tax deductible. Investment grows tax-free. Finally, withdrawals for qualified medical expenses are also tax-free. To fully utilizes the advantages, it is better to let the money stay and grow in HSA as long as possible. To achieve this, Do NOT use HSA money to cover the current medical expenses. Instead, save the receipts and ask for reimbursement in future year of withdrawal. Remarks: HSA does NOT have any CA state tax benefits. Both HSA enrollees and their spouse are NOT eligible to contribute to the general   flexible spending account (FSA) . One can have a HSA and  limited purpose FSA (LPFSA) at the same time. Unlike the general FSA, LPFSA can only cover vision and dental expenses. High Deduction v.s. HSA Saving To be eligible for HSA,
Read more