Stochastic Processes · 免费预览
收敛性与极限下的积分
随机变量的逐点收敛本身并不能保证期望随之收敛；有界收敛、单调收敛和支配收敛定理给出了使这种交换合法的三个标准假设。
1. A warning example
A warning example
2. Bounded convergence
Bounded convergence
3. Monotone convergence
Monotone convergence
4. Dominated convergence and the discipline of hypotheses
Dominated convergence and the discipline of hypotheses
5. Why the expectation stays put
In the warning example with P of the singleton k equal to 2 to the minus k, and X_n equal to 2 to the n on the point n and 0 elsewhere, why does E X_n fail to converge to the expectation of the pointwise limit?
6. Applying bounded convergence
A sequence of random variables satisfies the absolute value of X_n at most 3 for every n, and X_n converges to X pointwise. What does bounded convergence conclude?
7. Monotone convergence hypotheses
Which hypothesis is the defining requirement of the monotone convergence theorem, as stated in this lesson?
8. Dominated convergence needs a budget
Under dominated convergence, what exactly must the dominating variable Y satisfy?
9. Auditing the warning example
Which is the sharpest true statement about why all three convergence theorems fail on the warning example sequence?
10. A quick interchange
Random variables X_n all take values in the interval from 0 to 1, and X_n converges to X pointwise. Which conclusion is justified, and by which theorem?
11. The stubborn expectation
In the warning example, X_n equals 2 to the n on the single point n with probability 2 to the minus n and is 0 elsewhere. Compute E X_n and write the number.
12. The spike's footprint at n equal to 5
In the same warning example, compute the probability that X_5 is strictly positive and write the answer as a fraction or a decimal.
13. Missing the spike at n equal to 4
In the same warning example, compute the probability that X_4 equals 0 and write the answer as a fraction or a decimal.