Let X be a continuous random variable. What value of b minimizes E (|X-b|)? Giv

In summary, the value of b that minimizes the expected value of the absolute difference between a continuous random variable X and b can be found by taking the integral form of the expectation and differentiating it with respect to b. This approach may involve breaking up the integral into two parts if the absolute value sign causes complications.
  • #1
johnG2011
6
0
Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Giv

Homework Statement



Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation


The Attempt at a Solution



E(|X - b|)

E[e - [itex]\bar{x}[/itex]] = E(X)

E(|E[e - [itex]\bar{x}[/itex]] - b|)

so ?,... 0 = E(|E[e - [itex]\bar{x}[/itex]] - E|)

but this is a graduate course, I have a funny feeling that I am supposed to derive this using a the integral of an Expected value.
 
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  • #2


what is e? Its not clear what your steps are attempting

I think the integral would be a good way to approach this
 
  • #3


lanedance said:
what is e? Its not clear what your steps are attempting

I think the integral would be a good way to approach this

The e is supposed to be an observation in the sample set
 
  • #4


ok well its still not real clear what you're trying to do

i would try and write the expectation in integral form and consider differentiating, though you may need to be careful with the absolute value
[tex]f(b) = E[|X-b|] = \int_{-\infty}^{\infty} dx.p(x).|x-b| [/tex]
 
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  • #5


if the absolute value sign gives you trouble, you could consider using b to break up the integral into a sum of two integrals (x<b and x>b), this however will complicate the differentiation as now b appears in the integration limit also
 
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FAQ: Let X be a continuous random variable. What value of b minimizes E (|X-b|)? Giv

What is a continuous random variable?

A continuous random variable is a type of random variable in statistics that can take on any value within a certain range. This is different from a discrete random variable, which can only take on specific, separate values.

What does the term "minimizes" mean in this context?

In this context, "minimizes" means finding the value of b that makes the expected value of the absolute difference between X and b as small as possible. Essentially, it is finding the value of b that makes the expression E(|X-b|) as close to 0 as possible.

How is the value of b related to the random variable X?

The value of b is a constant that is subtracted from X in the expression E(|X-b|). This allows us to calculate the expected value of the absolute difference between X and b. The value of b is not dependent on the random variable X, but rather it is a value that we are trying to determine in order to minimize the expression.

How do you calculate the expected value of |X-b|?

The expected value of |X-b| can be calculated using the integral E(|X-b|) = ∫ |X-b| f(x) dx, where f(x) is the probability density function of the continuous random variable X. This integral can be solved using integration techniques, such as integration by parts or substitution.

What is the significance of finding the value of b that minimizes E(|X-b|)?

Finding the value of b that minimizes E(|X-b|) allows us to determine the value of X that is closest to the mean of the distribution. This can be useful in understanding the central tendency of the data and making predictions about future values of X. It is also a way to measure the spread or variability of the data around the mean.

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