Question about errors, Hubble's constant

In summary, the author is looking for a way to calculate the weighted average of two errors with correlations.
  • #1

Homework Statement


I am just looking through some old notes I have from for cosmology, and there's something cropped up that i can't seem to figure out:

Say I have two (or more) values for [itex]H_o[/itex] each with errors such as:

[tex]H_{o_1}=70^{+a+b}_{-c-d}[/tex]
and

[tex]H_{o_2}=69^{+e+f}_{-g-h}[/tex]

How would I go about calculating the weighted averaged (a,c,e,g are statistical errors. The rest are systematic errors) and then uncerstainty on the weighted average when for instance [itex]a\neq c[/itex].

Homework Equations


All the formula i found are along the lines of:

[tex]\bar{x}=(\sum^{N}_{i=1}x_i/\sigma_i^2)/(\sum^{N}_{i=1}1/\sigma_i^2)[/tex]

[tex]\sigma_{\bar{x}}=\sqrt{1/(\sum^{N}_{i=1}1/\sigma_i^2})[/tex]

The Attempt at a Solution


I've attempted to workout the top uncertainty on it's own, and likewise with the bottom but that doesn't seem the right way to do it.
 
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  • #2
To do it properly, you first have to know about the correlations between the systematic uncertainties. Then you can get the likelihood functions of the individual measurements, combine them, and then extract central value and uncertainties from that again.
If you just have access to the given numbers and expect that the correlation is small, the quick and dirty weighted average should give some reasonable approximation. The uncertainty of the weighted average follows from the usual uncertainty propagation.
 
  • #3
Ah okay thank you, I have just been given numbers and no correlation and been told to make an assumption. So I should say that if i assume the correlation between systematic uncertainties is small.

So to work that out, let's say I have:

[tex]76.9^{+3.9+10}_{-3.4-8}[/tex]
[tex]66^{+11+9}_{-10-8}[/tex]
How would I go about using that in the formula I have above for the weighted average. for instance what would I use for [itex]\sigma_1[/itex] when its values for the upper and lower uncertainties differ.
 
  • #4
I would probably use the average of the upwards and downwards uncertainty for the weights. If those the uncertainties are too asymmetric, this simplified approach will fail anyway.
 
  • #5
Okay thankyou! ill give it a try now, i did have attempt at doing each on their own it gave a strange result so ill try taking the average.
 

1. What is Hubble's Constant?

Hubble's Constant is a measure of the rate at which the universe is expanding. It is denoted by the symbol H0 and has units of kilometers per second per megaparsec (km/s/Mpc).

2. How is Hubble's Constant calculated?

Hubble's Constant is calculated by measuring the redshift of light from distant galaxies and using this information to determine their distance from us. This is then compared to the recessional velocity of the galaxies, which is measured through the Doppler effect. The ratio of these two values gives us Hubble's Constant.

3. What errors can affect measurements of Hubble's Constant?

There are several sources of error that can affect measurements of Hubble's Constant. These include uncertainties in the distance measurements to galaxies, variations in the recessional velocity of galaxies due to their individual motions, and potential systematic errors in the calibration of instruments used to measure redshift and distance.

4. Why is Hubble's Constant important in cosmology?

Hubble's Constant is important in cosmology because it helps us understand the history and evolution of the universe. By measuring the rate of expansion, we can determine the age and size of the universe, as well as gain insights into the nature of dark energy, a mysterious force believed to be responsible for the accelerating expansion of the universe.

5. How has our understanding of Hubble's Constant changed over time?

Our understanding of Hubble's Constant has evolved over time as new and more precise methods of measurement have been developed. Initially, Edwin Hubble estimated a value of 500 km/s/Mpc, while more recent measurements using the Hubble Space Telescope have refined this value to be around 70 km/s/Mpc. However, there is still ongoing research and debate in the scientific community about the exact value of Hubble's Constant.

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