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Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers

Instructions

  • This section covers the concepts listed below. 
  • For each concept, there is a conceptual video explaining it followed by videos working through examples. 
  • When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page
Tail Probability

Examples


Directions: There are no video examples for this topic.
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. What are the left and right tail probabilities for a random variable?
  2. Is expectation an increasing function? In what sense exactly?
Markov’s Inequality

Examples


Directions: The following examples cover the material from the video above. 
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. State Markov's Inequality.
  2. To which random variables can Markov's inequality be applied?
  3. What do you need to know in order to use Markov's Inequality?
Chebyshev’s Inequality

Examples


Directions: The following examples cover the material from the video above. 
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. State Chebyshev's Inequality.
  2. To which random variables can Chebyshev's Inequality be applied?
  3. What do you need to know in order to use Chebyshev's Inequality?
Convergence in Probability

Examples


Directions: The following examples cover the material from the video above. 
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. Define convergence of a sequence of random variables to a number.
Weak Law of Large Numbers

Examples


Directions: The following examples cover the material from the video above. 
 

Self-Assessment Questions


Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
  1. State the Weak Law of Large Numbers. Which inequality proves the Weak Law of Large Numbers?