Stochastic Processes · 免费预览
条件期望
给定事件的条件期望按该事件的概率对积分进行缩放，满足线性性并符合詹森不等式；对一个划分上的条件均值取平均即可恢复无条件均值，这正是构建鞅所依赖的工具。
1. Averaging over part of the world
Averaging over part of the world
2. Linearity and pulling out constants
Linearity and pulling out constants
3. Averaging over a partition
Averaging over a partition
4. Computing with the machinery
Computing with the machinery
5. The rescaled integral
How is the conditional expectation of Y given an event A defined?
6. The independence shortcut
When is E of Y given A simply equal to E of Y?
7. One identity to keep
What does the law of total expectation over a partition A 1 through A k say?
8. Constants on the piece
Suppose X equals 5 on the event A. What is E of X Y given A?
9. Convexity survives conditioning
What does Jensen's inequality say in conditional form?
10. The dice cross-check
The total of two dice is averaged conditionally on the first die's face, giving conditional means k plus 3.5. What does the partition formula return?
11. A six on the first die
Two fair dice are rolled and A is the event that the first shows 6. The second die is uniform from 1 to 6 with mean 3.5. Compute E of the total given A, and write the number.
12. Revenue by day of week
A shop averages 400 dollars on weekdays, which occur five sevenths of the time, and 900 on weekend days. Compute the unconditional expected daily revenue and write the answer as a fraction or decimal.
13. A pass or fail reward
A reward is 40 dollars if a test is passed and 10 if failed, and the test is passed with probability three quarters. Compute the expected reward and write the answer as a decimal.