For x and y binary digits (0 or 1), let x ⊕y be 0 if x=y and 1 if x ≠y (this operation is called

step 1 All right, so if we let Z be equal to X, exclusive, or Y, where X or equals 1, when the two digi

For x and y binary digits (0 or 1), let x ⊕y be 0 if x=y and...

For $x$ and $y$ binary digits $(0$ or 1$)$, let $x \oplus y$ be 0 if $x=y$ and 1 if $x \neq y$ (this operation is called exclusive or (often abbreviated to XOR), or addition mod 2). (a) Let $X \sim \operatorname{Bern}(p)$ and $Y \sim \operatorname{Bern}(1 / 2)$, independently. What is the distribution of $X \oplus Y ?$ (b) With notation as in (a), is $X \oplus Y$ independent of $X ?$ Is $X \oplus Y$ independent of $Y ?$ Be sure to consider both the case $p=1 / 2$ and the case $p \neq 1 / 2$. (c) Let $X_{1}, \ldots, X_{n}$ be i.i.d. Bern(1/2). For each nonempty subset $J$ of $\{1,2, \ldots, n\}$, let $$ Y_{J}=\bigoplus_{j \in J} X_{j} $$ where the notation means to "add" in the $\oplus$ sense all the elements of $J$; the order in which this is done doesn't matter since $x \oplus y=y \oplus x$ and $(x \oplus y) \oplus z=x \oplus(y \oplus z)$. Show that $Y_{J} \sim$ Bern $(1 / 2)$ and that these $2^{n}-1$ r.v.s are pairwise independent, but not independent. For example, we can use this to simulate 1023 pairwise independent

For $x$ and $y$ binary digits $(0$ or 1$)$, let $x \oplus y$ be 0 if $x=y$ and 1 if $x \neq y$ (this operation is called exclusive or (often abbreviated to XOR), or addition mod 2).
(a) Let $X \sim \operatorname{Bern}(p)$ and $Y \sim \operatorname{Bern}(1 / 2)$, independently. What is the distribution of $X \oplus Y ?$
(b) With notation as in (a), is $X \oplus Y$ independent of $X ?$ Is $X \oplus Y$ independent of $Y ?$ Be sure to consider both the case $p=1 / 2$ and the case $p \neq 1 / 2$.
(c) Let $X_{1}, \ldots, X_{n}$ be i.i.d. Bern(1/2). For each nonempty subset $J$ of $\{1,2, \ldots, n\}$, let
$$
Y_{J}=\bigoplus_{j \in J} X_{j}
$$
where the notation means to "add" in the $\oplus$ sense all the elements of $J$; the order in which this is done doesn't matter since $x \oplus y=y \oplus x$ and $(x \oplus y) \oplus z=x \oplus(y \oplus z)$. Show that $Y_{J} \sim$ Bern $(1 / 2)$ and that these $2^{n}-1$ r.v.s are pairwise independent, but not independent. For example, we can use this to simulate 1023 pairwise independent

Key Concepts

Modulo 2 Arithmetic

Modulo 2 arithmetic, also known as arithmetic in the field GF(2), is a system where the numbers wrap around after reaching 2. It is especially notable for its simplicity, as its operations—addition and multiplication—follow rules that are analogous to binary logic. In this context, the XOR operation represents addition modulo 2, facilitating the analysis of random binary variables and their combinations, and plays a critical role in various applications including error-detection, error-correction, and cryptographic algorithms.

Independence and Pairwise Independence

Independence in probability means that the joint distribution of two or more random variables factors into the product of their marginal distributions, implying that knowledge of one variable does not affect the distribution of the others. Pairwise independence is a weaker condition where every pair of random variables is independent, but the entire set may not be mutually independent. This distinction is important in probability theory because certain constructions allow for pairwise independent random variables (which can be useful in simulation and complexity theory) even though they fail full independence when all variables are considered jointly.

Bernoulli Distribution

The Bernoulli distribution is a discrete probability distribution for a random experiment that has exactly two possible outcomes, typically labeled as 0 and 1. The distribution is characterized by a single parameter, p, which represents the probability of one of the outcomes (usually 1), while the probability of the other outcome is 1 - p. This distribution is fundamental in probability theory and is used to model binary events such as coin flips, success/failure experiments, and other situations with two outcomes.

Exclusive Or (XOR) Operation

The exclusive or, or XOR, is a binary operation that takes two binary digits and returns 0 if the digits are the same and 1 if the digits are different. Often considered as addition modulo 2, this operation is commutative and associative, which makes it a useful algebraic tool in areas such as cryptography, coding theory, and computer science, as well as in probabilistic contexts where random binary variables are combined.

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