The content provided appears to be in Russian and concerns probability theory and random variables. Here are the answers to each of the questions based on the provided text:

430:

Given the probability density function $f(x) = \frac{1}{\pi(1 + x^2)}$, the question asks for the probability $P(0 \leq x \leq 1)$. This is the integral of the probability density function over the interval [0, 1]:

$P(0 \leq x \leq 1) = \int_0^1 \frac{1}{\pi(1 + x^2)} \, dx$

This integral is a standard arctangent integral, and the result is:

$P(0 \leq x \leq 1) = \frac{1}{2}$

Thus, the answer is 0.5.

431:

The question asks for the value of $F(+\infty)$, where $F(x)$ is the cumulative distribution function (CDF) of the random variable. For a probability distribution, the CDF at $+\infty$ is always 1, as the total probability must sum to 1.

Thus, the answer is 1.

432:

The question asks for the value of $F(-\infty)$. For most probability distributions, $F(-\infty) = 0$, as the probability of the random variable being less than $-\infty$ is zero.

Thus, the answer is 0.

433:

The question asks for the behavior of the probability density function $f(x)$. The options are:

• $f(x) < 0$
• $f(x) \leq 0$
• $f(x) \geq 0$
• $f(x) > 0$

Since the probability density function must be non-negative, the correct answer is:

$f(x) \geq 0$.

434:

This question relates to the derivative of the cumulative distribution function. If $f(x)$ is the probability density function, then the derivative of the cumulative distribution function $F(x)$ is equal to the probability density function:

$f(x) = \frac{d}{dx} F(x)$

Thus, the answer is:

$f(x) = F'(x)$.

435:

The question asks for the relationship between the CDF values for two different values $x_1$ and $x_2$, where $x_1 < x_2$. Since the cumulative distribution function is non-decreasing, we have:

$F(x_1) \leq F(x_2)$

Thus, the correct answer is:

$F(x_1) \leq F(x_2)$.

436:

This question asks about the integral of the probability density function for a continuous random variable $X$. The total probability must equal 1, and the integral of the probability density function over all possible values of $x$ must be 1:

$\int_{-\infty}^{+\infty} f(x) \, dx = 1$

Thus, the answer is:

$\int_{-\infty}^{+\infty} f(x) \, dx = 1$.

Let me know if you need further explanations or have more questions!

Related Questions:

1. How can I calculate the probability for any given interval using a probability density function?
2. What is the difference between a probability density function and a cumulative distribution function?
3. How do you compute the probability for continuous random variables?
4. What are the properties of cumulative distribution functions for random variables?
5. How do we apply integration in probability theory?

Tip: When dealing with probability density functions (PDFs), remember that the area under the curve (integral of the PDF) over any interval gives the probability of the random variable lying within that interval.