Example. Solve the Cauchy problem for the heat equation

\[
\begin{gathered}
u_{t}=u_{x x}, \quad x \in(-\infty,+\infty), \quad t \in(0,+\infty) \\
u(x, 0)=e^{-x} \quad x \in(-\infty,+\infty)
\end{gathered}
\]
Solution.

The solution to the Cauchy problem for the heat equation on a line is determined by the Poisson formula (3). For \(a=1, \alpha=0\) and \(\beta=-1\), we have

\[
u(x, t)=\frac{1}{2 \sqrt{\pi t}} \int_{-\infty}^{+\infty} e^{-\frac{(x-\xi)^{2}}{4 t}} e^{-\xi} d \xi
\]

Thus, the problem reduces to computing the integral (6).

1. We make a change of variable in the integral (6):

\[
\frac{\xi-x}{\sqrt{4 t}}=y, \quad \frac{d \xi}{\sqrt{4 t}}=d y, \quad \xi=2 \sqrt{t} y+x
\]

We obtain

\[
u(x, t)=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-y^{2}} \cdot e^{-2 \sqrt{t} y-x} d y
\]

2. Completing the square in the expression

\[
-y^{2}-2 \sqrt{t} y-x=-(y+\sqrt{t})^{2}+t-x
\]

we get

\[
u(x, t)=\frac{1}{\sqrt{\pi}} \int_{-\infty}^{+\infty} e^{-(y+\sqrt{t})^{2}} e^{t-x} d y=e^{t-x} \int_{-\infty}^{+\infty} e^{-(y+\sqrt{t})^{2}} d y=e^{t-x}
\]

Answer. \(u(x, t)=e^{t-x}\).

Problem conditions. Solve the Cauchy problem for the heat equation

1. \(u_{t}=2 u_{x x}, \quad x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-x^{2}-2 x}\).
2. \(u_{t}=3 u_{x x}, \quad x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-2 x^{2}+x}\).
3. \(u_{t}=4 u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-4 x^{2}-3 x}\).
4. \(u_{t}=u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-2 x^{2}+5 x}\).
5. \(u_{t}=3 u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-x^{2}+3 x}\).
6. \(u_{t}=5 u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-3 x^{2}-x}\).
7. \(u_{t}=2 u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty)\)

\(u(x, 0)=e^{-4 x^{2}+x}\)
8. \(u_{t}=u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty)\)

\(u(x, 0)=e^{-x^{2}+3 x}\).
9. \(u_{t}=4 u_{x x}\)

\(x \in(-\infty,+\infty), t \in(0,+\infty)\)

\(u(x, 0)=e^{-x^{2} / 2-x}\).
10. \(u_{t}=3 u_{x x}, \quad x \in(-\infty,+\infty), t \in(0,+\infty), \quad u(x, 0)=e^{-x^{2} / 3+2 x}\).

## Answers.

1. \(u(x, t)=\frac{1}{\sqrt{1+4 t}} e^{\frac{t(2 x-2)^{2}}{1+4 t}-x^{2}-2 x}\).
2. \(u(x, t)=\frac{1}{\sqrt{1+24 t}} e^{\frac{t(2 x-2)^{2}}{1+4 t}-x^{2}-2 x}\).
3. \(u(x, t)=\frac{1}{\sqrt{1+64 t}} e^{\frac{4 t(8 x-3)^{2}}{1+64 t}-4 x^{2}-3 x}\).
4. \(u(x, t)=\frac{1}{\sqrt{1+8 t}} e^{\frac{t(4 x+5)^{2}}{1+8 t}-2 x^{2}+5 x}\).
5. \(u(x, t)=\frac{1}{\sqrt{1+12 t}} e^{\frac{3 t(2 x+3)^{2}}{1+12 t}-x^{2}+3 x}\).
6. \(u(x, t)=\frac{1}{\sqrt{1+60 t}} e^{\frac{5 t(6 x-1)^{2}}{1+60 t}-3 x^{2}-x}\).
7. \(u(x, t)=\frac{1}{\sqrt{1+32 t}} e^{\frac{2 t(8 x+1)^{2}}{1+32 t}-4 x^{2}+x}\).
8. \(u(x, t)=\frac{1}{\sqrt{1+4 t}} e^{\frac{t(2 x+3)^{2}}{1+4 t}-x^{2}+3 x}\).
9. \(u(x, t)=\frac{1}{\sqrt{1+8 t}} e^{\frac{4 t(x-1)^{2}}{1+8 t}-\frac{1}{2} x^{2}-x}\).
10. \(u(x, t)=\frac{1}{\sqrt{1+4 t}} e^{\frac{3 t(2 x / 3+2)^{2}}{1+4 t}-\frac{1}{3} x^{2}+2 x}\).

## Chapter 6

## PROBABILITY THEORY

When studying the topic of PROBABILITY THEORY, you will be introduced to various examples of probabilistic models of random phenomena (experiments): the classical probability model, geometric probabilities, the Bernoulli trials scheme, and the simplest event flow. You will learn to compute the probabilities of random events (judgments about the results of a random experiment) using the probabilities of so-called elementary outcomes or using formulas that allow finding the probabilities of their combinations. In this process, you will become familiar with one of the most important concepts in probability theory - the concept of independent random events. You will learn to analyze random variables (generalized numerical characteristics of a random experiment), finding their distribution (probabilistic description) or some averaged characteristics, such as the expected value and variance. You will become familiar with the main distributions of random variables: Poisson, Cauchy, Laplace, uniform, normal, exponential, binomial, geometric, and others. You will learn to find the characteristic functions of random variables and from them - the expected value and variance. You will learn to analyze random vectors, finding their distribution or such averaged characteristics as the vector of scatter or the covariance matrix. You will learn to analyze numerical functions of random variables or vectors. Finally, you will learn to compute probabilities related to the sums of a large number of independent and identically distributed random variables, using the central limit theorem.

The package RESHEBNIK.VM will help you perform calculations and provide the necessary reference information. In particular, the package RESHEBNIK.VM can easily compute the values of standard functions often used in solving probabilistic problems.

## 6.1. Classical Probability Model

Problem Statement. A certain random experiment is conducted, in which the number of elementary outcomes is finite and all of them are equally likely (tossing several coins or dice, purchasing a certain number of lottery tickets, selecting several cards from a deck, etc.). Find the probability of a certain random event \(A\) associated with this experiment.

Solution Plan. When the number of equally likely elementary outcomes is finite, the probability of an event is the ratio of the number of elementary outcomes favorable to this event to the total number of elementary outcomes.

1. Define the space of elementary outcomes.

This is characterized by the fact that any realization of the random experiment can be described by one and only one elementary outcome.

2. Verify the equal likelihood of elementary outcomes.

In solving problems, the equal likelihood of elementary outcomes follows from considerations of symmetry or is assumed to be guaranteed by the methodology of conducting the random experiment.

In practice, the equal likelihood of elementary outcomes of a random experiment is verified by analyzing its results using mathematical statistics.

3. Select the elementary outcomes that lead to the occurrence of event \(A\) (favorable outcomes).

The criterion for selection is the description of event \(A\).

4. Count the total number of elementary outcomes \(N\) (the number of elements in the space of elementary outcomes).

This is done directly or using combinatorics.

5. Count the number of elementary outcomes \(N_{A}\) favorable to event \(A\).

This is done directly or using combinatorics.

6. According to the classical definition of probability

\[
\mathrm{P}(A)=\frac{N_{A}}{N}
\]