Inequalities and Limit Theorems
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Each question is graded on a single answer: a check that this chapter's methods work, not a test of the reasoning behind them.
A non-negative random variable has . What upper bound does Markov's inequality give for ?
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About these 105 questions
Markov, Chebyshev, Cantelli and Chernoff bounds; Jensen's inequality; the four modes of convergence and the implications between them; the weak and strong laws of large numbers; and the central limit theorem with Berry-Esseen and the delta method. This set covers the chapter Inequalities and Limit Theorems in Probability: 34 easy, 44 medium and 27 hard questions, each graded instantly with a hint and a full step-by-step solution. Progress is saved in this browser only.
- Markov, Chebyshev, and the price of assuming nothing — 18 questions
- Exponential tails: Chernoff and Hoeffding bounds — 17 questions
- Jensen's inequality — 15 questions
- Four ways a sequence of random variables can converge — 16 questions
- The laws of large numbers — 17 questions
- The central limit theorem — 22 questions
For work on paper there is a printable problem set with separate worked solutions.