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 \le 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.) What is the expected number of student that will fail this course if there are 32 enrolled?

b.) What is 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 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 increase by a factor of 4?

Half Width -vs- variance

Show by a numerical calculation that, for the Gaussian probability distribution, the full-width at half maximum [math](\Gamma)[/math] is related to the standard devision by [math]\Gamma = 2.354 \sigma[/math].

Gaussian Probability

a.) What is the probability of observing a value 1 standard deviation from the mean for a Gaussian distribution? ie: [math]P_G(\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, \mu, \sigma)[/math]

Lorentzian Area

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

Error Propagation

[1] Forest_Error_Analysis_for_the_Physical_Sciences