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EFR515 Statistical calculation课程简介

This course introduces the logic and methods of statistics. We begin with a discussion of the role of statistics, introducing the concepts of internal and external validity. Common methods for describing the characteristics of individuals and educational outcomes are presented, including the use of graphs and summary measures such as the mean, median, standard deviation, and correlation coefficient. Because of the large natural variability among individuals, one must be able to determine whether or not an apparent difference or patterns present in the data seems to be merely a chance occurrence.
Probability concepts are introduced to help us in this effort. Probability then forms the basis of all of the inferential statistical procedures subsequently presented. At the end of the course, students will:

• identify basic statistical applications for educational research.
• explain how to implement quantitative approaches to educational research.
• identify types of statistical methods and strategies, and select data collection and analysis approaches for different research interests.
• integrate fundamental statistical theories and concepts with functions of SPSS programs in the context of an analysis project’s overall design.

PREREQUISITES 

Students have 3 to 9 months to complete 18 lessons, including a final research project. Each lesson contains a variety of items which include required chapter reading, a quiz on the reading assignment, at least one instructor video and supporting material for the lesson, and a lesson activity assignment. Lesson topics include:

1. Introduction to statistics
2. Frequency Distributions
3. Central Tendency
4. Variability
5. Z scores: Standardized distributions
6. Probability
7. Probability Sampling
8. Hypothesis Testing
9. $t$ Statistic
10. Independent Sample $t$ Test
11. Related Samples $t$ Test
12. Intro to Analysis of Variance
13. Repeated Measure Analysis of Variance
14. Two-factor Analysis of Variance
15. Correlations
16. Regression
17. Chi-square
18. Research Project

EFR515 Statistical calculation HELP（EXAM HELP， ONLINE TUTOR）

The Fermi energy is the energy of the highest occupied quantum state in a system of non-interacting fermions at zero temperature. For a one-dimensional Fermi gas with spin-1/2 particles, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m} \left(\frac{\pi n}{2}\right)^2EF​=2mℏ2​(2πn​)2

where $n$ is the one-dimensional number density of fermions and $m$ is their mass. At zero temperature, all quantum states up to the Fermi level are occupied, and all states above the Fermi level are unoccupied. The total energy of the system is given by the sum of the energies of all occupied quantum states:

E = \int_{-\infty}^{E_F} g(E) E \, dEE=∫−∞EF​​g(E)EdE

where $g(E)$ is the density of states. For a one-dimensional gas, the density of states is given by:

g(E) = \frac{m}{\pi\hbar^2}g(E)=πℏ2m​

Substituting this expression into the equation for the total energy and integrating, we obtain:

E = \frac{\hbar^2}{4m} \left(\frac{\pi n}{2}\right)^2E=4mℏ2​(2πn​)2

We can see that the total energy is proportional to the Fermi energy, with a factor of $\frac{1}{2}$. This is a general property of non-interacting fermion systems in any dimension.

For a two-dimensional Fermi gas, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m} \left(\pi n\right)EF​=2mℏ2​(πn)

The density of states for a two-dimensional gas is given by:

g(E) = \frac{m}{\pi\hbar^2}g(E)=πℏ2m​

Substituting this expression into the equation for the total energy and integrating, we obtain:

E = \frac{\pi}{2} n E_F = \frac{\hbar^2}{4m} \left(\pi n\right)^2E=2π​nEF​=4mℏ2​(πn)2

Again, we can see that the total energy is proportional to the Fermi energy, with a factor of $\frac{\pi}{2}$.

For a spin-1/2 Fermi gas, the pressure is given by the Fermi gas equation of state:

P = \frac{2}{3} E \cdot n,P=32​E⋅n,

where $E$ is the total energy of the gas, $n$ is the number density of fermions, and we have used the fact that the kinetic energy of a Fermi gas is $3/5$ of its total energy.

Taking the partial derivative of $P$ with respect to $\rho$ at $T=0$ gives:

\left(\frac{\partial P}{\partial \rho}\right)_{T=0} = \frac{2}{3} n \left(\frac{\partial E}{\partial \rho}\right)_{T=0}(∂ρ∂P​)T=0​=32​n(∂ρ∂E​)T=0​

To find $\left(\frac{\partial E}{\partial \rho}\right)_{T=0}$, we use the fact that for a non-interacting Fermi gas, the Fermi energy is given by:

E_F = \frac{\hbar^2}{2m}(3\pi^2n)^{2/3}EF​=2mℏ2​(3π2n)2/3

At $T=0$, the energy per particle is given by:

\frac{E}{N} = \frac{3}{5} E_FNE​=53​EF​

Taking the derivative of this expression with respect to $n$ gives:

\left(\frac{\partial E}{\partial n}\right)_{T=0} = \frac{2}{3} E_F \cdot \frac{1}{n}(∂n∂E​)T=0​=32​EF​⋅n1​

Substituting this expression into the equation for $\left(\frac{\partial P}{\partial \rho}\right)_{T=0}$, we obtain:

\left(\frac{\partial P}{\partial \rho}\right)_{T=0} = \frac{2}{5}\frac{E_F}{m}(∂ρ∂P​)T=0​=52​mEF​​

The sound velocity $u$ is defined as:

u^2 = \left(\frac{\partial P}{\partial \rho}\right)_{T=0}u2=(∂ρ∂P​)T=0​

Substituting the expression we found above for $\left(\frac{\partial P}{\partial \rho}\right)_{T=0}$, we obtain:

u^2 = \frac{2}{5}\frac{E_F}{m} = \frac{2}{5} \frac{\hbar^2}{2m^2}(3\pi^2n)^{2/3}u2=52​mEF​​=52​2m2ℏ2​(3π2n)2/3

For a Bose gas below the Bose-Einstein temperature, the pressure is given by the ideal Bose gas equation of state:

P = \frac{2}{3} E \cdot nP=32​E⋅n

where $E$ is the total energy of the gas, $n$ is the number density of bosons. The energy of an ideal Bose gas is given by:

E = \frac{3}{2} N k_B T \zeta\left(\frac{5}{2}\right)\left(\frac{m}{2\pi \hbar^2}\right)^{3/2}E=23​NkB​Tζ(25​)(2πℏ2m​)3/2

where $N$ is the total number of bosons, $k_B$ is Boltzmann’s constant, $T$ is the temperature, $\zeta(x)$ is the Riemann zeta function, and we have used the fact that the Bose gas has a kinetic energy equal to $3/2$ of its total energy.

Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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