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Section 4: Expectation, Variance, and Continuous Random Variables

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
Expectation

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. What is the definition of \(E[X]\) for a discrete random variable, and what is its meaning?
  2. Does expectation always exist?
  3. What is the best you can say about \(E[X]\) if \(X\geq 0\)?
Variance and Standard Deviation

Examples


Directions: There are no example videos 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. Let \(X,Y\) be two random variables with \(E[X]=E[Y]\) and \(Var(X)<Var(Y)\). What does this tell you about \(X,Y\)?
  2. Define the moments of a random variable.
  3. Write standard deviation in terms of moments.
Properties of Expectation and Variance

Examples


Directions: There are no example videos for this topic.
Continuous Random Variable and Probability Density Function

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. What is the significance of probability density functions, and for which type of random variables are they useful?
  2. What is \(P(X=3)\) if \(X\) is a continuous random variable?
  3. How does the CDF determine whether the corresponding random variable is discrete, continuous, or mixed?
  4. How do you find the probability density function of a continuous random variable if we know its CDF and vice versa? 
Probability Density Function
 

There is no conceptual video for this topic.
 

Examples


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

Expectation of Continuous Random Variable

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 \(E[X]\) if \(X\) is a continuous random variable.
Uniform Probability Density Function

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. Is \(X^2\) uniformly distributed if \(X\) is a random number selected uniformly between 0 and 1?