Introduction to Probability · 免费预览
不等式：詹森、马尔可夫、切比雪夫与切尔诺夫
概率论中常用的不等式：用于期望的柯西–施瓦茨不等式和詹森不等式，用于尾界估计的马尔可夫不等式和切比雪夫不等式，以及用于获得指数级精确界的切尔诺夫不等式。
1. Expectation inequalities
Expectation inequalities
2. Tail inequalities
Tail inequalities
3. Chernoff and matching tools to information
Chernoff and matching tools to information
4. Jensen's direction
g is convex. Which statement is Jensen's inequality?
5. Choosing the right bound
You know only that a nonnegative variable has mean 4. Which bound can you place on P(X is at least 20)?
6. Chebyshev at k = 2
Chebyshev bounds P(absolute deviation from the mean at least 2 standard deviations) by 1/k squared. For k = 2, enter the bound as a decimal, like 0.25.
7. Markov's ceiling
A repair bill averages 300 dollars. Bound P(bill at least 1000). Enter a decimal like 0.3.
8. Jensen on the square root
E(X) = 16. Since the square root is concave, E of sqrt(X) is at most sqrt(16). Enter that ceiling as an integer.
9. In your own words
In your own words, state the most important definition, theorem or method of this lesson and explain what it says.