# Tag Info

23

No, one doesn't need to measure the material for years - or even millions or billions of years. It's enough to watch it for a few minutes (for time $t$) and count the number of atoms $\Delta N$ (convention: a positive number) that have decayed. The lifetime $T$ is calculated from $$\exp(-t/T) = \frac{N - \Delta N}{N}$$ where $N$ is the total number of atoms ...

20

I agree with @Ron Maimon that these ETS questions are problematic. But this is (i think) the reasoning they go with. Unlike @Mike's assumption you should not take the normal average, but as stated in the question the weighted average. A weighted average assigns to each measurement $x_i$ a weight $w_i$ and the average is then $$\frac{\sum_iw_ix_i}{\sum_i w_i}... 12 Actually a paper recently came out, and highlighted in Popular Science, discussing using fermionic field concepts to model crowd avoidance at Netflix. You can imagine that the same concept could be used to consider in any situation where there are large numbers of people competing for limited preferred items. Update Now that we have a few minutes, ... 12 Here is how I interpret what happened: You used Excel to compute the coefficients of the Gaussian that best describe the data: mean \mu, standard deviation \sigma, and magnitude A for a curve$$Y=Ae^{-(x-\mu)^2/2\sigma^2}$$Then you evaluated that function at a number of X values. Since the X values are not symmetrical about the calculated mean, you ... 11 Geant is a framework---which means that you use it to build applications that simulate the detector and physics you are interested in. The simulation can include all of physics and the complete detector including electronics and trigger (i.e. you can write your simulation so that it output a data file that looks just like the one you are going to get from ... 10 You might like this 110-page paper by me and Alex Arkhipov, which is all about a quantum bosonic analogue of Galton's board (we even use the same graphic you did -- see Section 1.1!). In particular, we gave strong evidence that such a board (with an arbitrary configuration of "pegs," and with multiple entry points for the "balls") is exponentially hard even ... 9 This is something that particle physicists are perfectly well aware of. For any given observed effect, there is always a nonzero probability that the observation will be a false positive that was caused by a random fluctuation. The name of the game is taking enough data that this probability is small enough. In general, the more data you take, the less ... 9 I see where you are going with your question. Let me feed the flames. The sigma value that is quoted is equivalent to a false alarm probability. It tells you how unlikely it is for your experiment, given your understanding (theoretically and empirically) of the noise characteristics, to have produced a signal that looked like GWs from a merging BH. ... 8 Typically one knows the functional form of something when they are interested in the FWHM. In this case, you can use least-squares fitting to fit the function to the data and extract the parameters. A very common functional form of a resonance is the Lorentzian:$$f(A,\gamma,x_0;x) = A\frac{1}{1 + \gamma^2\left(x-x_0\right)^2}$$where 1/\gamma gives the ... 8 If there are enough data and the prior is not completely unreasonable, the frequentist and the Bayesian approach give essentially the same answer. This is related to the central limit theorem. If data are fairly scarce, the two approaches may differ a lot. In this case the Bayesian approach is far preferable but only if the prior reflects true prior ... 8 The sign of a good fit is that the residuals have the same distribution as your model for the errors. Usually the assumption that goes into fitting methods is that the errors are normally distributed. That is, given perfect inputs x_i, and an ideal relation y_i = f(x_i), you will measure y_i + \epsilon_i, where \epsilon_i are distributed normally. ... 8 Half life is, by definition, the amount of time until half of an infinitely large sample would decay. That's precisely equivalent (according to the frequentist interpretation of probability, if that matters to you) to the time until an individual particle's probability of decay reaches one half. The half life is a theoretical quantity that doesn't depend on ... 7 The formula you've specified$$ \Delta k = \sqrt{(\Delta k_1)^2 + (\Delta k_2)^2} $$is the formula to obtain error of quantity k, as being dependent on k_1 and k_2 according to the following expression$$ k = k_1 + k_2.$$Generally, to obtain experimental error of a dependent quantity (and the expression stated in your question), you start with ... 7 The fact that the quantities \langle p^2\rangle and \langle p\rangle^2 are different is not something specific to quantum mechanics, but exists in any context where one can define an average value. Let's take a simple example and suppose that p is a balanced binary random variable which can take the values +1 and -1. Since the variable is balanced,... 7 Standard deviation adds uncertainties to the measured value: 23.3\pm 0.4\,{\rm m}. One can quickly look at the error (which has units of {\rm m} in my case) and think, The value could be as low as 22.9\,{\rm m} or as high as 23.7\,{\rm m} without much thinking. Modifying this to being a percentage of the value would be confusing. Plus it would be ... 7 No because none of them know the actual answer. The averaging process you describe only works if each estimate is of the exact answer plus noise. Otherwise it is known as the "Emperor's nose" problem. Nobody can see the Chinese emperor's face so they ask a million peasants how long his nose is, they average the results, and since they have such a large 'N' ... 7 No, of course not. Yes, some people will overestimate and others will underestimate. Averaging would cancel out the bias to some extent, but there's no reason to expect it to cancel out the bias perfectly. We all have similar eyes and brains. We are all deceived by the same optical illusions, in the same way. We all have a shared cultural understanding of ... 7 Model two consecutive measurements as the real values plus some noise. Call the first measured temperature T_1 and the second T_2. Call the measured noises \gamma_1 and \gamma_2, and suppose that they are drawn from a distribution \Gamma(\gamma) and are uncorrelated. The (approximation to the) derivative is$$\text{Derivative} \approx \frac{(T_2 +...

6

The main problem is to determine what corresponds to zero mass of the harmonic oscillator. Remember that a fraction of the spring mass also participates in the motion. By introducing an intercept $\beta$, your friend takes into account that the true zero of the mass parameter $m$ may be shifted from what you think it is. So an affine model $T^2=Cm+\beta$ is ...

6

Maybe for the same reason that experimental physics groups do not have a theoretician as a group member. One could think of experimental groups as ruled by "control freaks", they need as members experimetnalists who have mastered enough theory to set up the experiments and enough theory to interpret the results. Within this "theory" one could count ...

6

Errors in particle physics are of two kinds. Statistical, and systematic. Statistical is the usual standard deviation of gausian distributions, sqrt(n)/N for 1 sigma. It is the systematics that take a lot of effort, and often are not taken well into account. Systematic errors come from 1) the background to the signal expected. The background is calculated ...

6

I don't think there's any way you're going to do this in six months. I'll give a calculation below, but first an order of magnitude estimate. If you've detected a total of $N_{\rm events}$ events, your measurement of a modulation will have an error of order $N_{\rm events}^{-1/2}$ -- -- these things always do! -- so the number of events required is going ...

6

The ultimate answer is the JCGM 100:2008 guide followed by most of the metrology institutes around the world. The specific chapter on combining uncertainties is Chapter 5. Specifically, for a two-variable function $f(t_1, t_2)$ of two random variables, Eq. (16) of Section 5.2.2. gives $$\Delta f^2= \left (\frac{\partial f}{\partial t_1} \right )^2 \Delta ... 6 I think you're exercising an incorrect picture of statistics here - mixing the inputs and outputs. You are recording the result of a measurement, and the spread of these measurement values (we'll say they're normally distributed) is theoretically a consequence of all of the variation from all different sources. That is, every time you do it, the length of ... 6 Yes, the only sensible formula for the total error is the sum in quadrature,$$ \Delta X_{\rm total} = \sqrt { \Delta X_{\rm syst}^2 + \Delta X_{\rm stat}^2 } $$The key assumption behind the validity of the formula is that the two sources of error are independent i.e. uncorrelated.$$ \langle \Delta X_{\rm syst} \Delta X_{\rm stat} \rangle = 0 Because of ...

6

think about this with an example: the sine and cosine functions. They both average individually to zero over an interval. You can multiply those averages and still obtain zero. But if you multiply sin by itself and then average, you get a very distinct non-zero result. When the functions are arbitrary, the average of the product quantifies statistical ...

6

The quantity on the right side of the expression for the product of uncertainties basically depends on the mathematical definition of "uncertainty" one used. Without a rigid mathematical definition of this quantity one often just say that the product of uncertainties in position and momentum is of the order of Planck constant (or the reduced Planck constant; ...

6

I think that the easiest way to understand this is in the formula for addition. If you consider your quantity $C$ which depends on $A$ and $B$ such that $C=A+B$, then the formula $\Delta C = \Delta A + \Delta B$ overestimates the error values. You can visualize this as a rectangle with $A$ on the x-axis and $B$ on the y-axis with the area enclosed being \$\...

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