PHYS 114 Exam 1 Answer Key NAME:

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1 PHYS 4 Exam Answer Key AME: Please answer all of the questions below. Each part of each question is worth points, except question 5, which is worth 0 points.. Explain what the following MatLAB commands do as completely and succinctly as possible. These are all valid commands. a) n = hist(y,x) The variable n will contain the histogram (frequency of occurrence) of the vector Y evaluated at the bins given by the array x. The arrays n and x are the same size. The histogram will also be printed on the screen (because there is no line-ending semicolon). b) a = importdata('c:\windows\somethingsomething\phys4_data.txt'); The variable a will contain the data contained in the text file specified. The form and contents of a will depend on the structure and contents of the file. c) x = 0:5; y = poisspdf(x,5); plot(x,y,'+') This will plot the Poisson distribution (PDF) with mean μ = 5, for the range of counts 0-5, using the + symbol. d) x = -pi:pi; subplot(,,); plot(x,cos(x)); subplot(,,); plot(x,sin(x), r ); This will create two plots, one above the other, with the upper one a red curve representing sin(x) and the lower one a blue curve representing cos(x), for x = π to π. e) img = imread( 'C:\Windows\SomethingSomething\PHYS4_image.png'); image(img(:,:,)) colormap(gray(8)) This will display the green color plane of the input PG file, using a 8-value gray colormap. If the green color plane has values greater than 8, they will be clipped.

2 . Assume you have 4 marked, 6-sided dice and roll the collection simultaneously. a) How many different ways can you roll one? [List any assumptions!] To roll exactly one, there are four different ways, one for each die. The only assumption is that each die has one and only one marked on it. b) Assuming fair dice, what are the arguments (x, n and p) of P B (x; n, p) in this case? In this case, x =, n = 4, and p = /6. c) On any given roll of all 4 dice, what would be the probability of rolling three s? In this case, x = 3, and the probability is P B (x; n, p) = P B (3; 4, /6) = (4!/3!!)(/6) 3 (5/6) = 0/96 = d) Write the above answer in the form chance out of, where is the nearest appropriate integer. There is out of about 65 chance of rolling three s. 3. You are given the following measurements, A=[04, 90, 0, 5, 95]; which represent counts in a single pixel of a CCD camera. The integration time for each measurement is s. Find: a) The mean of the sample of five measurements. Write down the equation you used. The mean of the five measurements is The equation is x = x i

3 b) The variance of the sample of five measurements. Write down the equation you used. The variance of the five measurements is 6.7. The equation is s = ( xi x) c) ow, assume that you are told that there is a background bias noise of counts/s. What is the new SR over the 5 seconds of the total measurement? The new mean is just 04.8 = 93.8, while the variance is unchanged. The SR is mean/standard deviation, or mean/sqrt(variance), so it is 93.8/.75 = 5.. So SR = 5.. d) Assuming QE=73%, and.6 electrons/count, how many photons/s are needed to give the mean number of counts [after correction for the bias noise]? The camera measures photons with 73% efficiency, so each photon releases 0.73 electrons, and.6 electrons are needed for a count. Thus, for 93.8 counts/s, we require 93.8 counts/s*(.6 electrons/count)*(/0.73 photons/electron) = 45 photons/s. e) What Δt (integration time) is necessary for a SR of 00 [after correction for the bias noise]? A 5 s integration time gives 5. SR. We need an SR of 00, which is 00/5. = 6.58 times higher. The SR is proportional to sqrt(mean) for Poisson noise, so we need the mean to be 6.58 times higher, which is about 43 times longer integration time, or about 5 s. 4. State which PDF (Binomial, Poisson or Gaussian) best describes the following experiments and why. a) Measuring the length of an air track. Gaussian, because the measurements are continuous b) Recording cosmic ray detections Poisson, which is used for counting experiments c) Measuring occurrence of product defects on an assembly line. Binomial, which is used for yes/no measurements d) Measuring the weight of ball bearings in a manufacturing process. Gaussian, because the measurements are continuous

4 e) Determining the number of failures in a large-scale drug test. Binomial, which is used for yes/no measurements 5. An undergrad comes to you showing data from a recent experiment they did at CER, depicted in the figure below. List/discuss things that are good and things that need improvements. Please be specific! (0 points!) This is the data that I took. It cost ~$4 k per data point The table has labels, a title, and a figure caption, which are all good. It also calls attention to a feature in the plot, which is potentially good. The scaling of the plot is good. [m] otice this point My Data Fig. : Red curve is the fit. The problems are that the x axis label is not descriptive of that axis, and has no units. The y axis label has units, but no descriptive text to say what is being plotted. The caption does not say what the blue points are. The axis labels and axis numbers should be larger. 6. Short answer questions! Please answer as accurately and succinctly as possible. a) What is a stationary process? A stationary process is one in which the PDF (i.e. mean, standard deviation) do not change over the time the measurements are being made. b) What is a primary similarity between the binomial PDF and the Poisson PDF that makes them different from PDFs like the Gaussian or Cauchy (Lorentz) distributions? The binomial and Poisson PDFs are discrete (defined only for integer argument), while those of the Gaussian and Cauchy distributions are continuous. c) What is wrong in reporting the following measurements: 6.55 cm ±.375 mm The units are different between mean and error, and significant figures between the two are not compatible. 5. ± 0. There are no units given.

5 d) What is a forward model? A forward model is an analytical model that describes the data using parameters. e) A phenomenon that is said to be governed by Poisson statistics has a measured mean of 50 counts and standard deviation of 3 counts. Why would you question either the assumption or the results? For Poisson statistics, the standard deviation should be about equal to the square-root of the mean, so for a mean of 50 counts we expect a standard deviation of something like 7 counts. f) What criterion would you use to decide if the detection of a faint star on a CCD image is real (not just random noise), and why? The brightness of the star should be more than 3σ above the noise, at least. For random noise peaks, one expects a peak above 3σ to occur only about 0.3% of the time (99.7% probability to be below 3σ). g) Fill in the next row of the binomial coefficients. If this represents coin flips, how many coins does that row represent? This row represents 5 coins. h) If a Poisson PDF has a probability of 0.4 at x = 3, how many times would you expect to measure 3 counts in 50 measurement intervals? The probability of 0.4 is normalize to unity, meaning for measurement. To find out how many to expect for 50 measurements, just multiply by 50: 0.4*50 =.. I would expect to measure 3 counts in about of the measurements. i) What is the mean and standard deviation of a Gaussian distribution given by ( x 5) 8 PG ( x; μσ, ) = e? π One can read the mean and standard deviation directly from the analytical expression the mean is 5, and standard deviation is.

6 Equation Sheet (you may want to refer to some of these during the exam): x = xi s = ( xi x) μ = lim xi σ = lim ( xi μ) n! x n x PB ( x; n, p) = p ( p) μ = np σ = np( p) x!( n x)! x μ μ P ( x; μ) = e σ = μ x! x μ PG ( x; μσ, ) = exp σ π σ n f( x) = f( x ) P( x ) or f( x) = f( x) P( x) dx j= j j