Suppose that we flip a fair coin n times to obtain n random...

Suppose that we flip a fair coin $n$ times to obtain $n$ random bits. Consider all $m=\left(\begin{array}{l}n \\ 2\end{array}\right)$ pairs of these bits in some order. Let $Y_{i}$ be the exclusive-or of the $i$ th pair of bits, and let $Y=\sum_{i=1}^{m} Y_{i}$ be the number of $Y_{i}$ that equal 1 . (a) Show that each $Y_{i}$ is 0 with probability $1 / 2$ and 1 with probability $1 / 2 .$ (b) Show that the $Y_{t}$ are not mutually independent. (c) Show that the $Y_{t}$ satisfy the property that $\mathbf{E}\left[Y_{i} Y_{j}\right]=\mathbf{E}\left[Y_{i}\right] \mathbf{E}\left[Y_{j}\right]$. (d) Using Exercise 3.15, find $\operatorname{Var}[Y]$. (e) Using Chebyshev's inequality, prove a bound on $\operatorname{Pr}(|Y-\mathbf{E}[Y]| \geq n)$.

Suppose that we flip a fair coin $n$ times to obtain $n$ random bits. Consider all $m=\left(\begin{array}{l}n \\ 2\end{array}\right)$ pairs of these bits in some order. Let $Y_{i}$ be the exclusive-or of the $i$ th pair of bits, and let $Y=\sum_{i=1}^{m} Y_{i}$ be the number of $Y_{i}$ that equal 1 .
(a) Show that each $Y_{i}$ is 0 with probability $1 / 2$ and 1 with probability $1 / 2 .$
(b) Show that the $Y_{t}$ are not mutually independent.
(c) Show that the $Y_{t}$ satisfy the property that $\mathbf{E}\left[Y_{i} Y_{j}\right]=\mathbf{E}\left[Y_{i}\right] \mathbf{E}\left[Y_{j}\right]$.
(d) Using Exercise 3.15, find $\operatorname{Var}[Y]$.
(e) Using Chebyshev's inequality, prove a bound on $\operatorname{Pr}(|Y-\mathbf{E}[Y]| \geq n)$.

Key Concepts

Variance and Covariance

Variance measures the spread or variability of a random variable around its mean, while covariance quantifies the degree to which two variables change together. In analyzing sums of random variables, particularly when only pairwise independence is present, knowing the covariances helps in accurately computing the total variance using properties such as E[XY] = E[X]E[Y] for pairwise independent variables.

Chebyshev's Inequality

Chebyshev's inequality is a key result in probability theory that provides an upper bound on the probability that a random variable deviates from its mean by a specified amount. This inequality is applicable regardless of the underlying distribution, as long as the mean and variance are known, making it a powerful tool for deriving concentration bounds in a variety of contexts.

Expectation

Expectation, or the mean of a random variable, is a fundamental concept that summarizes the average outcome one would expect in repeated trials of a random experiment. Expectations are linear, which means that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent.

Pairwise Independence vs Mutual Independence

Pairwise independence means that any two random variables are independent, even if the collection as a whole is not fully mutually independent. This concept is important in probability because certain properties such as the product of expectations holding for any two variables can occur without full mutual independence, affecting how variances add up.

Exclusive OR (XOR) Operation

The exclusive OR, or XOR, is a binary operation that outputs true or 1 only when its two binary inputs are different. In probability problems, XOR is often used to combine independent binary outcomes, and understanding its properties is key to analyzing derived random variables and their distributions.

Indicator Random Variables

Indicator random variables are used to represent events that either occur or do not occur, typically taking on values 1 or 0 respectively. They provide a convenient way to model binary outcomes and simplify the calculation of expectations and variances in probability theory by reducing the problem to sums of 0s and 1s.

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