TF ErrAna Homework

Errors

Find 3 published examples of data which is wrongly represented.

Identify what is incorrect about it. What does it mean to be wrongly presented? A typical example is a political poll which does not identify the statistical uncertainty properly or at all.

Create a Histogram using ROOT

some commands that may interest you

root [1] TH1F *Hist1=new TH1F("Hist1","Hist1",50,-0.5,49.5);
root [2] Hist1->Fill(10);
root [3] Hist1->Draw();

You can use the above commands but you need to change the names and numbers above to receive credit. You must also add a title to the histogram which contains your full name. You will printout the histogram and hand it in with the above two problems.

Notice how the square rectangle in the histogram is centered at 10!

Notice that if you do the commands

root [2] Hist1->Fill(10);
root [3] Hist1->Draw();

the rectangle centered a 10 will reach the value of 2 on the vertical axis.

Two dice are rolled 20 times. Create a histogram to represent the 20 trials below

Trial	Value
1	8
2	10
3	9
4	5
5	9
6	6
7	5
8	6
9	3
10	9
11	8
12	5
13	8
14	10
15	8
16	11
17	12
18	6
19	7
20	8

Mean and SD

Electron radius

The probability that an electron is a distance [math]r[/math] from the center of the hydrogen atom

[math]P(r) = Cr^2 \exp^{-2 \frac{r}{R}}[/math]

Doing the integrals by hand (no tables) ,

a.)Find the value of C

b.) Find the mean electron radius and standard deviation for an electron in a hydrogen atom

Histograms by Hand

Given the following test scores from 40 students.

Trial	Value	Trial	Value	Trial	Value	Trial	Value
1	49	11	90	21	69	31	74
2	80	12	84	22	69	32	86
3	84	13	59	23	53	33	78
4	73	14	56	24	55	34	55
5	89	15	62	25	77	35	66
6	78	16	53	26	82	36	60
7	78	17	83	27	81	37	68
8	92	18	81	28	76	38	92
9	56	19	65	29	79	39	87
10	85	20	81	30	83	40	86

a.) calculate the mean and standard deviation

b.) construct a histogram by hand which has 10 bins centered on 10,20,...

c.) Use ROOT to construct a histogram. Compare the mean and RMS from ROOT with your result in part (a) above. What is the difference between the RMS report in the ROOT histogram and the standard deviation you calculated in part (a)?

Variance using Probability function

Given that

[math]\sigma^2 = \lim_{N \rightarrow \infty} \frac{1}{N}\sum_{j=1}^n \left [ \left (x_j - \mu \right)^2) \right ][/math]

Justify that

[math]\lim_{N \rightarrow \infty} \frac{1}{N}\sum_{j=1}^n \left [ \left (x_j - \mu \right)^2 \right ] = \lim_{N \rightarrow \infty} \left [ \frac{1}{N}\sum_{j=1}^n x_j^2\right ] - \mu^2[/math]

Note

The standard deviation ([math]\sigma[/math]) is the root mean square (RMS) of the deviations.

RMS = [math]\sqrt{\frac{1}{N}\sum_i^N x_i^2}[/math] so [math]\sigma = \mbox{RMS}(x_i -\mu)[/math]

Binomial Probability Distributions

1.)Evaluate the following (at least one by hand)

a.) [math]{6\choose 3}[/math]

b.) [math]{4\choose 2}[/math]

c.) [math]{10\choose 3}[/math]

d.) [math]{52\choose 4}[/math]

2.) Plot the binomial distribution P(x) for n=6 and p=1/2 from x=0 to 6.

[math]P(x) = {n\choose x}p^{x}q^{n-x} [/math]

3.) Given the probability distribution below for the sum of the point on a pair of dice

[math]P(x) =\left \{ {\frac{x-1}{36} \;\;\;\; 2 \le x \le 7 \atop \frac{13-x}{36} \;\;\; 7 \lt x \le 12} \right .[/math]

a.) find the mean

b.) find the standard deviation [math](\sigma)[/math]

4.) Prove that [math]\sigma^2 = npq[/math] for the Binomial distribution.

Poisson Prob Dist

Variance

Show that [math]\sigma^2 = \mu[/math] for the Poisson Distribution starting with the definition of variance.

Binomial/Poisson Statistic

The probability that a student will fail this course is 7.3%.

a.) Calculate by hand (i.e. without a computer/calculator) the expected number of students that will fail this course if there are 32 enrolled?

b.) Calculate by hand the probability that 5 or more will fail in one semester.

Deadtime

In a counting experiment it is possible for a detector to be "too busy" recording the effects of a detected particle that it is unable to measure another particle traversing the detector during the short time interval. "Dead time" is a measure of the time interval over which your detector is unable to make a measurement because it is currently making a measurement.

Assume that particle hit your detector at a rate of [math]1 \times 10^6[/math] particles/sec and that your detector has a deadtime of 200 ns [math](200 \times 10^{-9} sec)[/math]. The mean number of particles hitting the detector during this deadtime is [math]\mu = 0.2[/math]. The detector efficiency is defined as

[math]\epsilon = \frac{\mbox{average number of particles counted}}{\mbox{number of particles passing through the detector in 200 ns}}[/math]

a.) Find the efficiency of the detector assuming the process follows the Poisson distribution.

b.) Graph the efficiency as a function of the incident particle flux for rates between 0 and [math]10 \times 10^6[/math] particles/sec.

Gaussian Prob Dist

Counting experiment variance

a.)What is the standard deviation for a counting experiment with a mean [math]\mu[/math] = 100.

b.)What is the standard deviation if the mean number of counts is increased by a factor of 4?

Half Width -vs- variance

Show that the full-width at half maximum [math](\Gamma)[/math] is related to the standard devision by [math]\Gamma = 2.3548 \sigma[/math] for the Gaussian probability distribution. Begin with the definition that

[math]P_G\left (\mu + \frac{\Gamma}{2} \right ) = \frac{P_G(\mu)}{2}[/math]

Gaussian Probability

a.) Determine, by direct integration, the probability of observing a value beyond 1 standard deviation from the mean of a Gaussian distribution? ie: [math]P_G(X \le \mu - \sigma, \mu, \sigma)+ P_G(X \ge \mu + \sigma, \mu, \sigma)[/math] =?

b.) Find the value of [math]A[/math] such that [math]P_G(\mu + \sigma, \mu, \sigma) = A P_G(\mu,\mu,\sigma)[/math].

c.) repeat parts a.) and b.) above for [math]P_G(\mu + P.E., \mu, \sigma)[/math] and [math]P_G(\mu + \Gamma/2, \mu, \sigma)[/math]

[math]P_G(X =\mu + \sigma, \mu, \sigma)=0.24197 = 0.60653 P_G(X = \mu , \mu, \sigma)[/math]

[math]P_G(X =\mu + PE, \mu, \sigma)=0.3178 = 0.7965 P_G(X = \mu , \mu, \sigma)[/math]

[math]P_G(X =\mu - \Gamma/2, \mu, \sigma)=0.19947 = 0.5 P_G(X = \mu , \mu, \sigma)[/math]

Error Propagation

Lorentzian

a.)What fraction of the area of a Lorentzian curve is enclosed within the interval [math](\mu \pm \frac{3}{2} \Gamma)[/math]?

b.) The probability of observing a value from the Lorentzian distribution that is more than 2 half-widths ([math]\Gamma/2[/math]) from the mean.

Derivatives

Find the uncertainty [math]\sigma_x[/math] in [math]x[/math] as a function of the uncertainties[math] \sigma_u[/math] and [math]\sigma_v[/math] in [math]u[/math] and [math]v[/math] for the following functions:

a.) [math]x = \frac{1}{2(u-v)}[/math]

b.) [math]x= uv^2[/math]

c.) [math]x = u^2 + v^2[/math]

d.) [math]x = \frac{u-v}{u+v}[/math]

Snell's Law

Given Snell's Law

[math]n_1 \sin(\theta_1) = n_2 \sin(\theta_2)[/math]

Assume [math]n_1=1[/math] is known with absolute certainty and find [math]n_2[/math] and it's uncertainty when the following angles are measured

[math]\theta_1 = (22.03 \pm 0.02)^{\circ}[/math]

[math]\theta_2 = (14.45 \pm 0.2)^{\circ}[/math]

Linear Fit

2 Eq. 2 unknowns

Determine [math]x[/math] & [math]y[/math] for the system below

[math]7x + 8y =100[/math]

[math]2x-9y = 10[/math]

3 Eq. 3 unknowns

Given the system of 3 Equations and 3 Unkowns:

[math]a_{11}x + a_{12}y + a_{13}z = A[/math]

[math]a_{21}x + a_{22}y +a_{23}z= B[/math]

[math]a_{31}x + a_{32}y +a_{33}z= C[/math]

or in matrix form

[math]\left( \begin{array}{ccc} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right)\left( \begin{array}{c} x \\ y \\z\end{array} \right)= \left( \begin{array}{c} A \\ B \\C \end{array} \right) [/math]

a.)Express a the solution for x,y, and z in matrix form assuming the remaining terms in the above are known ([math]A,B,C[/math] and [math]a_{ij}[/math] are known)

b.) Find expressions for x,y, and z in terms of A,B,C when

[math]x + y -z = A[/math]

[math]z= B[/math]

[math]2x + y +2z= C[/math]

Linear Data Fit

A metal rod is placed with its ends in two different temperature baths. One bath is at zero degree Celsius and the other at 100. The temperature along the rod is measure and given in the table below.

Trial	Posiiton (cm)	Temperature (Celsius)
1	1.0	15.6
2	2.0	17.5
3	3.0	36.6
4	4.0	43.8
5	5.0	58.2
6	6.0	61.6
7	7.0	64.2
8	8.0	70.4
9	9.0	98.8

a.) Fit the above data assuming a linear function T = A + B X

assuming all of the data has the same uncertainty. Report the value of A & B and the uncertainty in the parameters.

b.) repeat part a.) above except this time assume each data point has an uncertainty given by the Poisson distribution as:

[math]\sigma_i^2 \approx T_i[/math].

Polynomial Least Squares Fit

Matrix Inversion

Use the Gauss-Jordan method to find the inverse matrix [math]\tilde{A}^{-1}[/math] given

[math] \tilde{A} = \begin{bmatrix} 2 & -1 & 0\\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}. [/math]

Poly Fit using ROOT

Hand in your source code and graphs for Lab 13

Polynomial Least Squares Fit

Perform a least squares fit to the data in Lab 14 using the Grid search method from Lab 16.

This will be a way to check your grid search algorithm against a known solution. Use the Temp -vs- Voltage data and the Linear function in Lab 13 above.

Hand in your source code in ROOT and a graph showing the linear fit with the Grid search parameters and the parameters found through Matrix inversion.

[1] Forest_Error_Analysis_for_the_Physical_Sciences