Download Discrete Mean and Variance under Transformations

Expected Value, Variance and Standard Deviation of Discrete Probability Distributions
Expected Value: E ( X )    x  P( X  x ) 
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Variance:  2  Var ( X )  E  x    
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Standard Deviation:    2  Var ( X )
For questions 1-4, you are welcome to create your own distributions and use a calculator for
developing and testing your conjectures. I have given you a simple distribution to start.
Example of a probability distribution:
x
1 2 3
1 1 1
P ( X  x)
2 3 6
1) Determine the expected value, variance and standard deviation of this distribution.
2) If you multiply all X values by a constant (dilation), does the same transformation occur to
E(X) and Var(X)?
3) If you increase all X values by a constant (translation), does the same transformation occur to
E(X) and Var(X)?
4) If you square all X values, does the same transformation affect E(X) and Var(X)?
5) How can these results be used to generate the following convenient expression for the
variance of a discrete probability distribution?
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Var ( X )  E  X 2    E  X   or Var ( X )  E  X 2    2
(In words: the variance of a discrete random variable is obtained by squaring the value of X and
obtaining the weighted mean of those squares, then by subtracting the square of the original
mean. Keep in mind that in order to obtain the standard deviation, you would have to find the
positive square root of this result.)
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Hint: start with Var ( X )  E  x    
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Reading after this assignment: 28D. Exercises: 28D.3 (1, 2, 3)
Some problems from Robert A. Powers
University of Northern Colorado