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1. Try to explain in your own words what is a random variable and what are its properties (explanations of following concepts are required: random variable, distribution of a random variable, realization of a random variable, mean, variance, histogram).
2. Generate graphs of probability distribution functions for the following distributions of random variable: normal, exponential, uniform (continuous) and Poisson. Describe what happens when you alter the parameters of aforementioned distributions.
3. From the data set attached to this task, generate histograms and try to determine the associated distributions.
4. Suppose we define a random variable $X$ as a result of a „fair“ (all outcomes are equally probable) six-sided dice roll. Determine the distribution function of the random variable $X$ and calculate $\mathrm {E} X$ and $\mathrm {var} X$.

Bonus: Name two different distributions of random variables with the same mean and variance.
For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

• script in the UTF-8 coding
• script in the CP-1250 coding
• data set

In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter very weakly so they can pass without any deceleration through Earth and similar large objects. Try to find out, using available literature and Internet sources, how many neutrinos are at any instant moment in an average person. Don't forget to reference the sources.

We are going up and down the same hill with the slope $α$, driving at the same speed $v$ and having the same gear (and therefore the same RPM of the engine), in a car with mass $M$. What is the difference between the power of the engine up the hill (propulsive power) and down the hill (breaking power)?

If we drop a bouncing ball or any other elastic ball on an appropriate surface, it starts to bounce. During every hit on the surface some kinetic energy of the ball is dissipated (into heat, sound, etc.) and the ball doesn't return to its initial height. We define the coefficient of restitution as the ratio of the kinetic energy after and before the hit. Is there any dependence between the coefficient of restitution and the height which the ball fell from? Choose one suitable ball and one suitable surface (or several if you want) for which you determine the relation between the coefficient of restitution and the height of the fall. Describe the experiment properly and perform a sufficient number of measurements.

Think up and describe a device that can deduce its orientation relative to gravitational acceleration and convert this information to an electrical signal. Come up with as many designs as you can. (An accelerometer-like device that is in most smart phones.)

With the aid of a digital camera measure the frequency of the AC voltage in the electrical grid. A smart phone with an app supporting manual shutter speed should be a sufficient tool.

We have a toilet paper roll with the diameter $R=8\;\mathrm{cm}$ with an inside hollow tube of diameter $r=2\;\mathrm{cm}$. Every layer of the paper has the thickness $d=200µm$ and the layers lies perfectly on top of each other. By how many does the number of pieces of the paper differ had we used a piece of the length $l_{1}=9\;\mathrm{cm}$ instead of $l_{2}=13\;\mathrm{cm}?$ A part of the solution has to be an estimate of the approximation error (if you use one).

Bonus: Calculate the precise length of the spiral the toilet paper makes.

Measure the tensile strenght of office paper. Use a common office paper with the density 80 g\cdot m^{−2}.

How tall could be a tower built from aluminium cans of diet soft drink?

Mage Greyhald celebrated his 100th birthday a long time ago and has begun to fear that Death will pay him a long delayed visit. He decided thus that he will ecase himself into a magic chest, where Death can't reach him. Unfortuntely he forgot to tell the craftsmen to add breathing holes. Air in the chest takes up a volume;$V_{0}=400l$, the percentage of the volume that is oxygen is $φ_{0}=0,21$. With every breath he uses up only $k=20%$ of the volume's oxygen $V_{d}=0,5l$. The frequency of breaths of the mage after the closing of the chest rises according to the relation

$$\\f(t)=f_0 \cdot \frac{\varphi_0}{\varphi (t)}\,,$ wheref_{0}=15breaths\cdot min^{−1}is$ the initial brath frequency is $φ(t)$ and the percetage of the volume that is oxygen at time case $t$. Determine how long until Death will come for Greyhald if the minimum volume of oxygen in the air required for survival is $φ_{s}=0,06$.