Show that $S_n$ and $X_{n+1}$ are independent.

Exercise VI

$\Omega$ denotes the sample space of a random experiment E and P denotes a probability on $\Omega$. $A$ and $B$ are two events with probabilities $0.6$ and $0.4$ respectively. We further assume that $P ( A \cup B ) = 0.8$. VI-A- $\quad P ( A \cap B ) = 0.24$. VI-B- $\quad A$ and $B$ are complementary events. VI-C- $\quad A$ and $B$ are independent events. VI-D- $\quad A$ and $B$ are mutually exclusive events.
For each statement, indicate whether it is TRUE or FALSE.

Let $S$ and $T$ be two independent random variables each taking a finite number of real values. Assume that $T$ and $- T$ follow the same distribution.
Show that:
$$E ( \cos ( S + T ) ) = E ( \cos ( S ) ) E ( \cos ( T ) )$$

Let $k$ be an integer greater than or equal to 2. We say that random variables $(Y_n)_{n \geq 1}$ are $k$-independent if $$\mathbb{E}\left[\psi_1(Y_{n_1}) \cdots \psi_k(Y_{n_k})\right] = \mathbb{E}\left[\psi_1(Y_{n_1})\right] \cdots \mathbb{E}\left[\psi_k(Y_{n_k})\right]$$ for all indices $1 \leq n_1 < \cdots < n_k$ and for all bounded functions $\psi_1, \ldots, \psi_k : \mathbb{R} \rightarrow \mathbb{R}$.
I.4.a) Prove that $k$-independence implies $j$-independence if $j \leq k$.
I.4.b) What is $N$-independence for $N$ random variables $Y_1, \ldots, Y_N$?
I.4.c) Let $Y_1$ and $Y_2$ be independent random variables with uniform distribution on $\{0,1\}$: for $n = 1,2$, $$\mathbb{P}(Y_n = 0) = \mathbb{P}(Y_n = 1) = \frac{1}{2}$$ Let $Y_3$ be the random variable on $\{0,1\}$ defined by $$Y_3 := Y_1 + Y_2 \quad \bmod 2$$ Prove that the random variables $(Y_1, Y_2, Y_3)$ are 2-independent but not 3-independent.

Let $k$ be an even integer in $\{2, \ldots, N\}$. We assume in this question that the random variables $X_1, \ldots, X_N$ are $k$-independent.
We introduce the following notations: $\mathcal{T}$ denotes the set $\{1, \ldots, N\}^k$. If $T = (n_1, \ldots, n_k) \in \mathcal{T}$ and $n \in \{1, \ldots, N\}$, we denote by $m_T(n)$ the multiplicity of $n$ in $T$, that is $$m_T(n) = \operatorname{Card}\left\{i \in \{1, \ldots, k\}; n_i = n\right\}$$ For $\ell \in \{1, \ldots, k\}$, we denote by $\mathcal{T}_\ell$ the set of $T$ in $\mathcal{T}$ involving exactly $\ell$ distinct indices, where each has multiplicity at least 2, namely: $T \in \mathcal{T}_\ell$ if $$\operatorname{Card}\left(\left\{n \in \{1, \ldots, N\}; m_T(n) > 0\right\}\right) = \ell,$$ and $$\forall n \in \{1, \ldots, N\}, \quad m_T(n) > 0 \Rightarrow m_T(n) \geq 2$$ Finally, we denote by $|\mathcal{T}_\ell|$ the cardinality of $\mathcal{T}_\ell$.
I.5.a) Determine $|\mathcal{T}_1|$ and $|\mathcal{T}_\ell|$ for $\ell > k/2$.
I.5.b) Justify $$\mathbb{E}\left[(S_N)^k\right] = \sum_{T \in \mathcal{T}} \prod_{n=1}^N \mathbb{E}\left[X_n^{m_T(n)}\right]$$ then $$\mathbb{E}\left[(S_N)^k\right] = \sum_{\ell=1}^{k/2} \sum_{T \in \mathcal{T}_\ell} \prod_{n=1}^N \mathbb{E}\left[X_n^{m_T(n)}\right]$$
I.5.c) Prove that $$\mathbb{E}\left[(S_N)^k\right] \leq \sum_{\ell=1}^{k/2} K^{k-2\ell} |\mathcal{T}_\ell|$$
I.5.d) Let $\ell \in \{1, \ldots, k/2\}$. Justify the following estimate: $$|\mathcal{T}_\ell| \leq \binom{N}{\ell} \ell^k \leq \frac{N^\ell}{\ell!} \ell^k$$ One may consider the set of $T \in \mathcal{T}$ involving at most $\ell$ distinct elements.
I.5.e) For $\ell \in \{1, \ldots, k/2\}$, prove that $$\ell! \geq \ell^\ell e^{-\ell}$$ then deduce that $$|\mathcal{T}_\ell| \leq (Ne)^\ell \left(\frac{k}{2}\right)^{k-\ell}$$
I.5.f) Prove that $$\mathbb{E}\left[(S_N)^k\right] \leq \left(\frac{Kk}{2}\right)^k \sum_{\ell=1}^{k/2} \left(\frac{2Ne}{kK^2}\right)^\ell$$
I.5.g) We assume $$kK^2 \leq N.$$ Prove that $$\mathbb{E}\left[(S_N)^k\right] \leq \frac{\theta}{\theta - 1} \left(\frac{Nek}{2}\right)^{k/2} \leq 2\left(\frac{Nek}{2}\right)^{k/2},$$ where $$\theta := \frac{2Ne}{kK^2}$$
I.5.h) Prove (under hypothesis (27)) the following estimate: for all $t > 0$, $$\mathbb{P}\left(|S_N| \geq t\sqrt{N}\right) \leq 2\left(\frac{\sqrt{ek/2}}{t}\right)^k$$

146- Is the relation $ad = bc \Leftrightarrow (a,b)\,R\,(c,d)$ an equivalence relation on $\mathbb{R}^2$? If it is an equivalence relation, which point does the graph $[(2,6)]$ pass through?
(1) It is not an equivalence relation. $(1,2)$ (2) $(1,3)$ (3) $(2,3)$ (4)

139. Let A and B be two independent events. If $P(A \cap B) = 0.6$ and $P(A \cap B) = 0.2$, then $P(A \cup B')$ is equal to:
(1) $0.7$ (2) $0.75$ (3) $0.85$ (4) $0.9$

10. If A and B are two independent events, prove that $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } ) . \mathrm { P } \left( \mathrm { A } ^ { \prime } \cap \mathrm { B } ^ { \prime } \right) \leq \mathrm { P } ( \mathrm { C } )$, where C is an event defined that exactly one of A and B occurs.
Sol. $P ( A \cup B ) . P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right) \leq ( P ( A ) + P ( B ) ) P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \cdot \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { A } ) ) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { B } ) )$ $\leq \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) = \mathrm { P } ( \mathrm { C } )$.

10. If A and B are two independent events, prove that $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } ) . \mathrm { P } \left( \mathrm { A } ^ { \prime } \cap \mathrm { B } ^ { \prime } \right) \leq \mathrm { P } ( \mathrm { C } )$, where C is an event defined that exactly one of A and B occurs.
Sol. $P ( A \cup B ) . P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right) \leq ( P ( A ) + P ( B ) ) P \left( A ^ { \prime } \right) P \left( B ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \cdot \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right)$ $= \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { A } ) ) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) ( 1 - \mathrm { P } ( \mathrm { B } ) )$ $\leq \mathrm { P } ( \mathrm { A } ) \mathrm { P } \left( \mathrm { B } ^ { \prime } \right) + \mathrm { P } ( \mathrm { B } ) \mathrm { P } \left( \mathrm { A } ^ { \prime } \right) = \mathrm { P } ( \mathrm { C } )$.

An experiment has 10 equally likely outcomes. Let $A$ and $B$ be two non-empty events of the experiment. If $A$ consists of 4 outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is
(A) 2, 4 or 8
(B) 3, 6 or 9
(C) 4 or 8
(D) 5 or 10

Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \leq |B| < |A|$, equals\_\_\_\_

Let $X$ and $Y$ are two events such that $P ( X \cup Y ) = P ( X \cap Y )$. Statement 1: $P ( X \cap Y' ) = P ( X' \cap Y ) = 0$. Statement 2: $P ( X ) + P ( Y ) = 2 P ( X \cap Y )$
(1) Statement 1 is false, Statement 2 is true.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is true, Statement 2 is false.
(4) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

Let $A$ and $B$ be two events such that $P ( \overline { A \cup B } ) = \frac { 1 } { 6 } , P ( A \cap B ) = \frac { 1 } { 4 }$ and $P ( \bar { A } ) = \frac { 1 } { 4 }$, where $\bar { A }$ stands for the complement of the event $A$. Then the events $A$ and $B$ are
(1) Independent but not equally likely.
(2) Independent and equally likely.
(3) Mutually exclusive and independent.
(4) Equally likely but not independent.

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of dice $A$ and $B$ is odd, then which of the following statements is NOT true? (1) $E_1$ and $E_3$ are independent. (2) $E_1$, $E_2$ and $E_3$ are independent. (3) $E_1$ and $E_2$ are independent. (4) $E_2$ and $E_3$ are independent.

Three persons $\mathrm { P } , \mathrm { Q }$ and R independently try to hit a target. If the probabilities of their hitting the target are $\frac { 3 } { 4 } , \frac { 1 } { 2 }$ and $\frac { 5 } { 8 }$ respectively, then the probability that the target is hit by P or Q but not by R is:
(1) $\frac { 39 } { 64 }$
(2) $\frac { 21 } { 64 }$
(3) $\frac { 9 } { 64 }$
(4) $\frac { 15 } { 64 }$

Let a set $A = A _ { 1 } \cup A _ { 2 } \cup \ldots \cup A _ { k }$, where $A _ { i } \cap A _ { j } = \phi$ for $i \neq j ; 1 \leq i , j \leq k$. Define the relation $R$ from $A$ to $A$ by $R = \left\{ ( x , y ) : y \in A _ { i } \right.$ if and only if $\left. x \in A _ { i } , 1 \leq i \leq k \right\}$. Then, $R$ is:
(1) reflexive, symmetric but not transitive
(2) reflexive, transitive but not symmetric
(3) reflexive but not symmetric and transitive
(4) an equivalence relation

Let $A = \{ 1,2,3 \}$. The number of relations on $A$, containing $( 1,2 )$ and $( 2,3 )$, which are reflexive and transitive but not symmetric, is $\_\_\_\_$

Let $\mathrm { A } = \{ 1,2,3 , \ldots , 9 \} ; \mathrm { xRy }$ iff $\mathrm { x } - \mathrm { y }$ is multiple of 3 . $5 _ { 1 } : $ Number of elements in R is 36 $\mathcal { S } _ { 2 } : R$ is equivalence relation (A) $\mathrm { S } _ { 1 } \& \mathrm {~S} _ { 2 }$ both are correct (B) $\mathrm { S } _ { 1 }$ is correct but $\mathrm { S } _ { 2 }$ is not correct (C) $\mathrm { S } _ { 1 } \& \mathrm {~S} _ { 2 }$ both are incorrect (D) $\mathrm { S } _ { 2 }$ is correct but $\mathrm { S } _ { 1 }$ is not correct

Let $\mathrm { M } = \{ 1,2,3 , \ldots , 16 \}$ and R be a relation on M defined by xRy if and only if $4 y = 5 x - 3$. Then, the number of elements required to added in R to make it symmetric is
(A) 3
(B) 2
(C) 5
(D) 4

For $\mathbf { H } , \mathbf { I }$ in question (1), and for $\mathbf { J }$, $\mathbf { K }$ in question (2), choose the appropriate answer from among (0) $\sim$ (3) at the bottom of this page.
For $\mathbf { L } \sim \mathbf { R }$ in question (3), enter the appropriate number.
Consider the following three possible conditions on two real numbers $x$ and $y$:
$p : x$ and $y$ satisfy the equation $( x + y ) ^ { 2 } = a \left( x ^ { 2 } + y ^ { 2 } \right) + b x y$, where $a$ and $b$ are real constants.
$$\begin{aligned} & q : x = 0 \text { and } y = 0 . \\ & r : x = 0 \text { or } y = 0 . \end{aligned}$$
(1) Suppose that in condition $p$, $a = b = 1$. Then $p$ is $\mathbf { H }$ for $q$, and $p$ is $\mathbf { I }$ for $r$.
(2) Suppose that in condition $p$, $a = b = 2$. Then $p$ is $\mathbf { J }$ for $q$, and $p$ is $\mathbf { K }$ for $r$.
(3) If in condition $p$ we set $a = 2$, we can transform the equation in $p$ into
$$\left( x + \frac { b - \mathbf { L } } { \mathbf { L } } y \right) ^ { 2 } + \left( \mathbf { N } - \frac { ( b - \mathbf { Q} ) ^ { 2 } } { \mathbf { P } } \right) y ^ { 2 } = 0 .$$
Hence $p$ is a necessary and sufficient condition for $q$ if and only if $b$ satisfies $\mathbf { Q } < b < \mathbf { R }$.
(0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition

Given two events, A and B, of a random experiment, with probabilities such that $p ( A ) = \frac { 4 } { 9 } , \quad p ( B ) = \frac { 1 } { 2 } y p ( A \cup B ) = \frac { 2 } { 3 }$, it is requested:\ a) (1 point) Check whether events A and B are independent or not.\ b) (1 point) Calculate $p ( \bar { A } / B )$, where $\bar { A }$ denotes the complementary event of A .

In a random experiment there are two independent events $X$, $Y$. We know that $P(X) = 0.4$ and that $P(X \cap \bar{Y}) = 0.08$ (where $\bar{Y}$ is the complementary event of $Y$). Find:\ a) (1 point) Calculate $P(Y)$.\ b) (0.5 points) Calculate $P(X \cup Y)$.\ c) (1 point) If $X$ is an undesired outcome, so that we consider the experiment a success when $X$ does NOT occur, and we repeat the experiment on 8 occasions, find the probability of having succeeded at least 2 times.

An event A has probability $\mathrm { P } ( \mathrm { A } ) = 0.3$.\ a) ( 0.75 points) An event B with probability $\mathrm { P } ( \mathrm { B } ) = 0.5$ is independent of A. Calculate $\mathrm { P } ( \mathrm { A } \cup \mathrm { B } )$.\ b) ( 0.75 points) Another event C satisfies $P ( C \mid A ) = 0.5$. Determine $P ( A \cap \bar { C } )$.\ c) (1 point) If there is an event D such that $P ( \bar { A } \mid D ) = 0.2$ and $P ( D \mid A ) = 0.5$, calculate $\mathrm { P } ( \mathrm { D } )$.

In a sample space there are two mutually exclusive events, $A _ { 1 }$ with probability 0.5 and $A _ { 2 }$ with probability 0.3, and $A _ { 3 } = \overline { A _ { 1 } \cup A _ { 2 } }$ is considered. Of a certain event $B$ with probability 0.4 it is known that it is independent of $A _ { 1 }$ and that the probability of the event $A _ { 3 } \cap B$ is 0.1 . With this data it is requested:
a) ( 1 point) Calculate the probability of $A _ { 3 }$.
b) ( 1.5 points) Decide whether $B$ and $A _ { 2 }$ are independent.

3

Consider a point P moving on the coordinate plane every 1 second according to the following rules (i), (ii).

• [(i)] Initially, P is at the point $(2,\ 1)$.
• [(ii)] When P is at point $(a,\ b)$ at some moment, 1 second later P is at
  • the point symmetric to $(a,\ b)$ with respect to the $x$-axis, with probability $\dfrac{1}{3}$
  • the point symmetric to $(a,\ b)$ with respect to the $y$-axis, with probability $\dfrac{1}{3}$
  • the point symmetric to $(a,\ b)$ with respect to the line $y = x$, with probability $\dfrac{1}{6}$
  • the point symmetric to $(a,\ b)$ with respect to the line $y = -x$, with probability $\dfrac{1}{6}$

Answer the following questions. For question (1), it suffices to state only the conclusion.

• [(1)] Find all possible coordinates of points that P can occupy.
• [(2)] Let $n$ be a positive integer. Show that the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start equals the probability that P is at point $(-2,\ -1)$ after $n$ seconds from the start.
• [(3)] Let $n$ be a positive integer. Find the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start.

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