1 / 16

More fun with PDF’s

More fun with PDF’s. How can we tell if something is changing?. Using PDFs & CLT, we can justify many of the things we have done to date. why sum errors (or other random noise) with square root of sum of squares? How can we tell if a measurement agrees with theory?

fredericka-hunter
Download Presentation

More fun with PDF’s

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. More fun with PDF’s How can we tell if something is changing?

  2. Using PDFs & CLT, we can justify many of the things we have done to date. • why sum errors (or other random noise) with square root of sum of squares? • How can we tell if a measurement agrees with theory? • Why do we assume 68% of time answer lies with in one standard deviation? • Why is the means the best estimate of a quantity if you have multiple measurements of it with error?

  3. I have stated that if we add the errors (or other random variables) x+y, with the STD x and y the sum has the variance • Why does this matter? • because we used it to predict the error in the mean!

  4. What is uncertainty in mean? • Remember from a few classes ago that the sum N random variables T is expected to scale as • And of course, the sum of N copies of T is NT, so the mean and its uncertainty is…. • Where T is the standard deviation of T • This is a fundamental result!

  5. How do we find the STD of the sum of two random, Gaussian, variables x and y with STD of x and y? • Assume <x>=<y>=0, for convenience. • Explain these distributions.

  6. What is the chance of getting a particular x and a particular y? • If z=x+y, what is the probability that we observe a paticular value of z? • use y=z-x to write the above as:

  7. If z=x+y, what is: • The probability that we observe a given value of x and z at the same time! • What do we want? P(z) !

  8. How do we get P(z) from P(x,z)? • Integrate over all possible x! (why?) • from table! • What does this mean?!? • Give result on board.

  9. Ok, that was a little stiff. • I tried to find an easier derivation…

  10. Next question. Assume we have either: • Two measurements, x±x and y ±y • Or one measurement x ±x and a theoretical value for x • And we want to see if they are significantly different. • We can never prove they are the same! • For example, was the mean temperature in fall of 2005 significantly different from the fall of 2006? • Or does your estimate of the mean rainfall in Durham match those of the weather service?

  11. We examine if the measurement xobs ±x is different from theoretical value for x, xtheory • Why? Because if we want to see if x±x and y ±y are different, we can make it into the above problem! • does x-y differ from zero? • the STD of x-y is sqrt(x2+ y2) • Bad joke; mathematician and engineer.

  12. Define three quantities: • xobs, what we have observed. We think the standard error in the observation is x. • xtheory, what we want to compare xobs to. • Assume that the error is Gaussian. • We want to know: What is the chance that xobs is significantly different from xtheory given the error x?

  13. first, we calculate “t” (why is it called t? no good reason!) t=(xobs-xtheory)/ x • where x is the standard deviation of the error in our estimate, xobs. • Why?

  14. 95% Don’t Trust 3 68% • Explain on board… Cheat sheet on next page!

  15. What is likelihood the real xobs is within ±t*x of the observed xobs? • if t=1, 68% within, 32% outside • if t=2, 95% inside, 5% outside (really t=1.96) • Most scientist use 5% chance limit, so significantly differrent if t>=1.96

More Related