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## statistical physics

### (6 points)3. Series 31. Year - 3. IDKFA

You fired at Imp from your plasma gun which shoots a cluster of particles with uniform velocity distribution in interval $\langle v_0, \; v_0+\delta v\rangle$ (all of the particles are moving in one line, there is no transverse velocity). The total kinetic energy of the cluster is $E_0$. The barrel rifle has a cross cestion of $S$ and the pulse takes an infinitely short time. How far does Imp need to stand to be safe. Assume that his skin is able to cool the heat flow of $q$.

### (3 points)0. Series 31. Year - 2.

### (10 points)6. Series 30. Year - S. nonlinear

- Describe in your own words how and when the nonlinear regression can be used (it is sufficient to describe the following: nonlinear regression model, estimation of unknown regression coefficients, expression of uncertainty of estimates of regression coefficients and fitted values, statistical tests of hypotheses about regression coefficients, identifiability of regression coefficients and choice of regression function). It is not necessary to provide derivations and proofs, brief overview is sufficient.

- See values ($x_{i},y_{i})$ in the attached file
*regrese1.csv*. We want to fit the theoretical functional dependence, which in this case is a sinusoid, i.e. the function

$$f(x)=a b\cdot \sin(cx d)$$

Plot a graph of observed values and fitted function (such graph has to meet usual requirements) and provide brief interpretation. It is not necessary to do regression diagnostics.

**Hint:** Do not forget to correctly solve the identifiability problem of this model by suitable restrictive conditions on possible values of parameter $c$.

- See values ($x_{i},y_{i})$ in the attached file
*regrese2.csv*. We want to fit the theoretical functional dependence, which, in this case, is an exponential function, i.e. the function

$$f(x)=a \;\mathrm{e}^{bx c}$$

Provide values of estimates of all regression coefficients including corresponding standard errors.

**Hint:** Try to check (by the means of graphical methods) whether the assumption of homoskedasticity holds and if necessary, use White's estimate (sandwich estimate) of covariance matrix to compute standard errors correctly.

- See values ($x_{i},y_{i})$ in the attached file
*regrese3.csv*. We want to fit the theoretical functional dependence, which in this case is a hyperbolic function, i.e. the function

$$f(x)=a \frac{1}{bx c}$$

Plot a graph of observed values (in the form of error bars) and fitted function and provide brief interpretation. Perform regression diagnostics.

**Bonus:** See values ($x_{i},y_{i})$ in the attached file*regrese4.csv*. We want to fit the theoretical functional dependence, which in this case is too complicated to be expressed in analytical form. Try to fit regression splines (with suitably chosen knots and suitable degree). Plot a graph of observed values and fitted function.

It is recommended to use statistical software *R* for all computations. Sample *R* script (with comments in code explaining syntax of \emph{R} programming language) may be helpful (in Czech only). .

sample script in UTF-8 encoding (in Czech only)

sample script in CP-1250 encoding (in Czech only)

ukazka1.csv (data file used in sample script)

Michal thinks that the last round has to be as difficult as possible.

### (9 points)4. Series 30. Year - P. statistician's daily bread

We've all been there, you spread some honey or some preserve on a slice of bread, take a bite, and suddenly, the spread drips through a hole and lands right on your hand. Determine how does the probability that there is a hole straight through a slice of bread depend on its thickness. The model of how does the dough rise is left up to you. (For example, evenly distributed bubbles with an exponential distribution of radii is a good model).

Michal stained his clothes.

### (2 points)5. Series 29. Year - 2. multiparticular

Let's have a container that is split by imaginary plane into two disjunct parts A and B, identical in size. There are $nparticles$ in the container and each of them has a probability of 50 % to be in part A and probability 50 % to be in part B. Figure out the probabilities of the part A containing $n_{A}=0.6n$ or $n_{A}=1+n⁄2$ particles respectively.. Solve it for $n=10$ and $n=N_{A}$, where $N_{A}≈6\cdot 10^{23}$ is Avogadro's constant.