Special Topics: Inequalities, Geometrical Probabilities, Difference Equations

If $g(x) \geq 0$ for every $x$ and $g(x) \geq c$ for $x \in (\alpha , \beta )$, then

(Continuation of Problem 270.) Show that for every constant $t > 0$

If $X_{1}$ and $X_{2}$ are independent and identically distributed random variables then for every $t > 0$

Cantelli's inequality. One tail Chebyshev's inequality. If $\mathbb {E}\left[X\right] = \mu$, $\operatorname {Var}\left[X\right] = \sigma^{2}$, then

(see Exercise 271).

Gauss inequality. If $x_{0}$ is the mode of a continuous unimodal distribution and $\tau^{2} = \mathbb {E}\left[(X-x_{0})^{2}\right]$ then

If $g(x) \geq 0$ and even, i.e., $g(x) = g(-x)$ and in addition $g(x)$ is nondecreasing for $x > 0$, show that for every $c > 0$

(Continuation of Problem 275.) If $g(x)$ of Exercise 275 satisfies $\left|g(x)\right| \leq M < \infty$, then

(Continuation of Problem 275.) Let $g$ be as in Exercise 275 and $\mathbb {P}\left(\left|X\right| \leq M\right) = 1$ then

Berge's inequality. Let $\mathbb {E}\left[X_{i}\right] = \mu_{i}$, $\operatorname {Var}\left[X_{i}\right] = \sigma_{i}^{2}$ $(i = 1, 2)$ and $\rho (x_{1}, x_{2}) = \rho$, then

Jensen's inequality for symmetric matrix functions (Cacoullos and Olkin, Biometrika, 1965). The symmetric matrix function $G(X)$ of the matrix $X$ of order $m \times n$ is called convex if, for every pair $X_{1}, X_{2}$ of $m \times n$ matrices and $0 \leq \lambda \leq 1$, by analogy to

(the defining inequality of an ordinary convex function $g$ on an interval $(\alpha , \beta )$), it satisfies

where for symmetric matrices $X, Y$ we write $X \geq Y$ if $X - Y$ is a nonnegative definite matrix. Let $X$ be a random matrix, i.e., a matrix whose elements are random variables and $G$ a convex matrix function. Then $G(\mathbb {E}\left[X\right]) \leq \mathbb {E}\left[G(X)\right]$.

For a positive random matrix show that

Show that if $\mathbb {E}\left[Y\right] < \infty$ and $Y > 0$ then

For independent and identically distributed random variables $X_{1}, X_{2}, \ldots , X_{n}$ with $\mathbb {P}\left(X_{i} > 0\right) = 1$ and $\operatorname {Var}\left[\log X_{i}\right] = \sigma^{2}$ show that for every $\varepsilon > 0$

Hence deduce

A point is chosen at random on each of two consecutive sides of a rectangle. What is the probability that the area of the triangle bounded by the two sides of the rectangle and the line segment connecting the two points is less than $3/8$ of the area of the rectangle?

Two points are selected at random on segment $AB$ which is thus divided into three parts. What is the probability that the three line segments can form a triangle?

What is the probability that three points selected at random on the circumference of a circle lie on a semicircle?

A line segment of length $1$ is divided into five parts by four points selected at random on it. What is the probability that each part is less than $1/2$?

A line segment $AB$ is divided by a point $C$ into two parts $AC = a$ and $CB = b$. Two points $X$ and $Y$ are taken on $AC$ and $CB$, respectively. What is the probability that $AX, XY, BY$ can form a triangle?

Buffon's needle problem. A smooth table is given with equidistant parallel lines at distance $2a$. A needle of length $2\mu$ $(\mu < a)$ is dropped onto the table. What is the probability that the needle crosses a line?

A closed convex curve with diameter less than $2a$ is dropped onto the table of Exercise 288. Find the probability that a line is crossed by the curve.

$A$ has $a$ 10-drachma coins and $B$ has $b$ 5-drachma coins. Each of them takes a coin out of his pocket and gives it to the other. This is repeated $n$ times. What is the probability that $A$ gives a 10-drachma coin to $B$ on the $n$th exchange?

$A$ tosses a die until an ace or a six appears twice.

A professor has three lecture rooms $A$, $B$, and $C$ available each year for teaching a course. Since he wishes to use a different classroom every year, at the end of the year he chooses one of the other two rooms for the next academic year as follows. The first year he teaches in $A$ and selects between $B$ and $C$ by tossing a coin. If heads appears he chooses $B$, otherwise he selects $C$. At the end of each year he makes his selection in the same way. Find the probability $p_{n}$ that he teaches in $A$ during the $n$th year. Similarly for $B$ and $C$.

Find the probability $p_{n}$ that in a sequence of $n$ Bernoulli trials, with probability of success $(S)$ $p$ the pattern $SF$ does not appear at all ($F \equiv$ failure).

Let $y_{k}$ be the expected number of occupied cells when placing (at random) $k$ balls into $n$ cells. Find $y_{k}$:

A pair of dice is thrown $n$ times; at each throw we observe whether a double six appears or not. Show that the probability $p_{n}$ of an even number of double six's satisfies the difference equation

Hence find $p_{n}$.

Each of $n$ urns contains $w$ white balls and $b$ black balls. One ball is transferred from the first urn to the second urn, then another ball from the second urn is transferred to the third one, and so on. Finally, we draw a ball from the $n$th urn. What is the probability that it will be white?

$A$ and $B$ play the following game. $A$ throws a coin I with probability of heads $p$ until he gets tails; then $B$ begins throwing another coin II with probability of heads $p'$ until he gets tails. Then $A$ begins again and so on. Given that $A$ begins the game, what is the probability of heads at the $n$th throw?

$A$ throws two dice and he gets $r$ points if $r$ aces appear $(r = 0, 1, 2)$; the game stops when no ace appears for the first time. Find the probability $p_{n}$ that he has $n$ points at stopping. If the game continues indefinitely, what is the probability that $A$ has $n$ points at some point of the game?

A box contains $a$ good lamps and $b$ bad lamps. At each draw of a lamp, if it is good it is kept in the box, otherwise it is replaced by a good one (from another box). What is the expected number of good lamps in the box after $n$ trials? Hence calculate the probability of selecting a good lamp at the $n$th trial.

Let $p_{k}$ denote the probability that no cell takes more than two balls when placing $k$ balls in $n$ cells at random. Find,