Mean of L2 norm of random vector Ax+n

In summary, the expected value of ||Ax+n|| where || || is the L2 norm and x and n are uncorrelated and E[n] = 0 is E[x'A'xA].
  • #1
cutesteph
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Homework Statement



What is the expected value of ||Ax+n|| where || || is the L2 norm and x and n are uncorrelated and E[n] = 0


The Attempt at a Solution


E[ norm of Y] = E[(Ax+n)' (Ax+n)] = E[(x'A'+n')(Ax+n)] = E[x'A'xA +x'T'n +n'Ax +n'n]
the three last terms = 0 due to uncorrelatedness so = E[x'A'xA] = E[tr(x'A'xA)] = E[tr(x'xA'A)] = A'AE[x'x] does this reduce any further?
 
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  • #2
cutesteph said:
E[ norm of Y] = E[(Ax+n)' (Ax+n)] = E[(x'A'+n')(Ax+n)] = E[x'A'xA +x'T'n +n'Ax +n'n]
You mean E[x'A'Ax +x'A'n +n'Ax +n'n] , right?
 
  • #3
Yes. But the result is still the same since it is a scalar and we can take the trace of it to rearrange. So does A'AE[x'x] simplify any further?
 
  • #4
cutesteph said:
Yes. But the result is still the same since it is a scalar
x'A'Ax is a scalar; x'A'x is a scalar; (x'A'x)A is an n x n matrix; x'(A'xA) is meaningless (since xA is meaningless).
 

FAQ: Mean of L2 norm of random vector Ax+n

1. What is the definition of mean of random vector?

The mean of a random vector is a statistical measure that represents the central tendency or average value of a set of random variables. It is calculated by summing up all the values in the vector and dividing by the total number of values.

2. How is the mean of random vector different from other measures of central tendency?

The mean of a random vector takes into account all the values in the vector, whereas other measures of central tendency such as median and mode only consider the middle or most frequent value. This makes the mean more sensitive to extreme values in the vector.

3. Can the mean of random vector be negative?

Yes, the mean of a random vector can be negative if there are negative values present in the vector. The mean is simply the average of all the values, regardless of their sign.

4. How is the mean of random vector calculated for a continuous distribution?

For a continuous distribution, the mean of a random vector is calculated using integration. This involves multiplying each value in the vector by its corresponding probability and summing them up to get the expected value.

5. What is the significance of the mean of random vector in data analysis?

The mean of a random vector is an important measure in data analysis as it provides a sense of the central tendency of the data. It can also be used to compare different datasets and make inferences about the population from which the random vector was sampled.

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