The value of E( (||X-μ||-c)^2 )

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In summary, the conversation discusses finding the value of E((||X- μ||-c)^2) for a normal distribution, given the values of X and μ. The conversation also mentions using the definition of expected value to solve this problem.
  • #1
saintman4
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I would like to ask one more question.

X ~ N (μ, σ2)

If X = [x1 x2] and μ = [μ1 μ2]. What is the value of E( (||X- μ||-c)^2 )?

where c is constant and E(||X- μ||^2)= σ2


Thank ...
 
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  • #2
[tex]\mathbb{E}[ (||X- \mu||-c)^2 ]=\mathbb{E}[ ||X- \mu||^2-2c||X-\mu||+c^2 ][/tex]

Now [itex]\mathbb{E}[\cdot][/itex] is linear ... does this help?
 
  • #3
Thank Pere Callahan. But I still don't know how to find E(||X- μ||) as I ask in https://www.physicsforums.com/showthread.php?t=224947 [Broken]

Do you know how to find E(||X- μ||)?

Thank you.
 
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  • #4
Did yu try to use the defintion of the expected value of some function g of a real random Variable X with density function f?
[tex]
\mathbb{E}[g(X)]=\int_{\mathbb{R}}{g(x)f(x)dx}
[/tex]

You will have to solve some Gaussian-type integrals but it should be straight-forward :smile:
 

1. What does "E( (||X-μ||-c)^2 )" represent?

E( (||X-μ||-c)^2 ) is a mathematical expression that represents the expected value of the squared difference between a random variable X and its mean μ, minus a constant c.

2. Why is this value important in scientific research?

This value is important because it measures the variability or dispersion of a dataset around its mean. It is often used in statistical analysis to assess the goodness of fit of a model or to compare different datasets.

3. How is this value calculated?

This value is calculated by taking the squared difference between each data point and the mean, then finding the average of all these squared differences. The constant c is then subtracted from this value. This process is known as calculating the mean squared error (MSE).

4. What does a higher or lower value of E( (||X-μ||-c)^2 ) indicate?

A higher value of E( (||X-μ||-c)^2 ) indicates that the data points are more spread out from the mean, while a lower value indicates that the data points are more tightly clustered around the mean. In other words, a higher value suggests a larger amount of variability or dispersion within the dataset.

5. Are there any limitations to using E( (||X-μ||-c)^2 ) as a measure of variability?

Yes, there are some limitations to using this value as a measure of variability. It assumes that the dataset follows a normal distribution, and it can be sensitive to outliers or extreme values in the dataset. Additionally, it does not provide information about the direction of the variability, only the magnitude. Therefore, it is important to consider other measures of variability and assess the overall distribution of the data when analyzing a dataset.

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