Probability Distributions

Let $F(x)$ be the distribution function corresponding to the probability distribution $\mathbb {P}$. Prove the equalities:

1. $\mathbb {P}\left([a, b]\right) = F(b)-F(a-)$;

2. $\mathbb {P}\left((a, b)\right)=F(b-)-F(a)$;

3. $\mathbb {P}\left([a, b)\right)=F(b-)-F(a-)$;

4. $\mathbb {P}\left(\left\{ x\right\} \right)=F(x)-F(x-)$.

Show that each of the functions

where [$\cdot$] is the integer part of a number, is right-continuous, increasing in each variable, but is not a distribution function on $\mathbb {R}^{2}$.

Let $\lambda_{1}, \ldots , \lambda_{n}$ be nonnegative numbers with $\lambda_{1}+\ldots +\lambda_{n}=1$. Let, moreover, $\mathbb {P}_{1}, \ldots , \mathbb {P}_{n}$ be probability distributions on $\left(\mathbb {R}^k, \mathscr {B}\left(\mathbb {R}^k\right)\right)$, and $F_{1}, \ldots , F_{n}$ the corresponding distribution functions. Is it true that the function $\lambda_{1} F_{1}+\ldots + \lambda_{n} F_{n}$ is a distribution function? If so, for which probability distribution?

Let $\lambda_{1}, \ldots , \lambda_{n}$ be nonnegative numbers with $\lambda_{1}+\ldots +\lambda_{n} = 1$. Let, moreover, $\mathbb {P}_{1}, \ldots , \mathbb {P}_{n}$ be absolutely continuous probability distributions on $\left(\mathbb {R}^k, \mathscr {B}\left(\mathbb {R}^k\right)\right)$, and $p_{1}, \ldots , p_{n}$ the corresponding densities. Is it true that the function $\lambda_{1} p_{1}+\ldots +\lambda_{n} p_{n}$ is a density? If so, for which probability distribution?

The density of the absolutely continuous distribution $\mathbb {P}$, given on $(\mathbb {R}, \mathscr {B}\left(\mathbb {R}\right))$, equals $p(x)$. Find the distribution function, if

1. $p(x)=\frac{\theta }{\pi \left(\theta^{2}+\left(x-x_{0}\right)^{2}\right)}$ (the Cauchy distribution with parameter $\theta$ and shift $x_{0}$);

2. $p(x)=\frac{1}{b-a}\; \mathbb {1}_{a \leq x \leq b}$ (the uniform distribution on $[a, b]$);

3. $p(x) = k(x-1)^{k-1} \; \mathbb {1}_{1 \leq x \leq 2},\; k \in \mathbb {N}$;

4. $p(x)=x e^{-x} \; \mathbb {1}_{x \geq 0}$ (the gamma distribution with parameters 2,1).

Let $\mathbb {P}$ be a discrete probability distribution on $(\mathbb {R}, \mathscr {B}\left(\mathbb {R}\right))$, $p(x)=\mathbb {P}\left(\left\{ x\right\} \right)$.

1. Find the distribution function corresponding to the probability distribution $\mathbb {P}$, if $p(x)=\frac{1}{2 N}\; \mathbb {1}_{x \in A}$, where

   $$A=\left\{ 1, \ldots , N\right\} \cup \left\{ 2 N+1, \ldots , 3 N\right\}$$

   (the uniform distribution on the set $A$).

2. Find $\mathbb {P}\left(2\mathbb {Z}_+\right)$, where $2 \mathbb {Z}_{+}$ is the set of nonnegative even numbers, if

   $$p(x)=\frac{\lambda ^{x} e^{-\lambda }}{x!} \; \mathbb {1}_{x \in \mathbb {Z}_+}$$

   where $\lambda >0$ (the Poisson distribution with parameter $\lambda$).

3. Find the distribution function corresponding to the probability distribution $\mathbb {P}$, and $\mathbb {P}\left(2\mathbb {Z}_+\right)$, if

   $$p(x)=p(1-p)^{x-1} \; \mathbb {1}_{x \in \mathbb {N}}$$

   where $p \in (0,1)$ (the geometric distribution with parameter $p$).

A meteorological station, when forecasting temperature, proceeds from the assumption that the temperature distribution is exponential with parameter 1 and with shift $\theta$, where the value $\theta$ depends on the month in which the forecast is made. Let $\theta_{1}=0$ be the temperature shift in April, and $\theta_{2}=5$ in May. Find the probability that the total temperature over one measurement in April and one in May does not exceed the number $c=15$, if the measurements are made independently.

Remark. The problem can be reformulated as follows. Let $\mathbb {P}$ be the probability measure on $(\mathbb {R}^{2}, \mathscr {B}\left(\mathbb {R}^{2}\right))$, defined by the equality $\mathbb {P}=\mathbb {P}_{1} \times \mathbb {P}_{2}$, where the density of the distribution $\mathbb {P}_{1}$ equals $e^{-\left(x-\theta_{1}\right)} I\left(x \geq \theta_{1}\right)$, and the density of the distribution $\mathbb {P}_{2}$ equals $e^{-\left(x-\theta_{2}\right)} I\left(x \geq \theta_{2}\right)$. One needs to find $\mathbb {P}\left(\left\{ (x, y) : x+y \leq 15\right\} \right)$.

Let $\mathbb {P}$ be a probability measure on $\left(\mathbb {R}^{3}, \mathcal{B}\left(\mathbb {R}^{3}\right)\right)$, defined by the equality $\mathbb {P}=\mathbb {P}_{1} \times \mathbb {P}_{2} \times \mathbb {P}_{3}$, where $\mathbb {P}_{1}$ and $\mathbb {P}_{2}$ are uniform distributions on $[0,1]$, and $\mathbb {P}_{3}$ is the exponential distribution with parameter $\lambda >0$. Find

1. $\mathbb {P}\left(\left\{ (x, y, z): x+y+z \leq 3\right\} \right)$;

2. $\mathbb {P}\left(\left\{ (x, y, z): x-y+z \geq 0\right\} \right)$;

3. $\mathbb {P}\left(\left\{ (x, y, z): 1 / 2 \leq x y \leq 3 z\right\} \right)$.

A shooter at a shooting range fires into a "quarter circle", i.e. into the region

The probability distribution of a hit $\mathbb {P}$ is uniform on the region $D$. In other words, the density of this distribution equals

Find

1. the distribution function and density of the marginal probability distribution $\mathbb {P}_{1}$, equal to the projection of $\mathbb {P}$ onto the first coordinate;

2. the probability that the shooter hits the square $\left[0, \frac{3}{4}\right] \times \left[0, \frac{3}{4}\right]$;

3. the probability of hitting the segment $\left[\frac{1}{2}, \frac{3}{4}\right]$ along the $y$-axis.

Let $\mathbb {P}$ be a probability measure on $\left(\mathbb {R}^{2}, \mathscr {B}\left(\mathbb {R}^{2}\right)\right)$, defined by the equality $\mathbb {P}=\mathbb {P}_{1} \times \mathbb {P}_{2}$, where $\mathbb {P}_{1}$ is the exponential distribution with parameter $\lambda_{1}>0$, and $\mathbb {P}_{2}$ is the Poisson distribution with parameter $\lambda_{2}>0$. Find

1. $\mathbb {P}\left(\left\{ (x, y): x+y \leq 1\right\} \right)$

2. $\mathbb {P}\left(\left\{ (x, y): x / y \geq 2\right\} \right)$.