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# 数学代写|数论代写Number Theory代考|STAT7604 Measures of randomness and the leftover hash lemma

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## 数学代写|数论代写Number Theory代考|Measures of randomness and the leftover hash lemma

In this section, we discuss different ways to measure “how random” a probability distribution is, and relations among them. Consider a distribution defined on a finite sample space $\mathcal{V}$. In some sense, the “most random” distribution on $\mathcal{V}$ is the uniform distribution, while the least random would be a “point mass” distribution, that is, a distribution where one point $v \in \mathcal{V}$ in the sample space has probability 1 , and all other points have probability 0.
We define three measures of randomness. Let $X$ be a random variable taking values on a set $\mathcal{V}$ of size $N$.

We say $X$ is $\delta$-uniform on $\mathcal{V}$ if the statistical distance between $X$ and the uniform distribution on $\mathcal{V}$ is equal to $\delta$; that is,
$$\delta=\frac{1}{2} \sum_{v \in \mathcal{V}}|\mathrm{P}[X=v]-1 / N| .$$

The guessing probability $\gamma(X)$ of $X$ is defined to be
$$\gamma(X):=\max {\mathrm{P}[X=v]: v \in \mathcal{V}} .$$

The collision probability $\kappa(X)$ of $X$ is defined to be
$$\kappa(X):=\sum_{v \in \mathcal{V}} \mathrm{P}[X=v]^2 .$$

## 数学代写|数论代写Number Theory代考|Basic definitions

To say that the sample space $\mathcal{U}$ is countably infinite simply means that there is a bijection $f$ from the set of positive integers onto $\mathcal{U}$; thus, we can enumerate the elements of $\mathcal{U}$ as $u_1, u_2, u_3, \ldots$, where $u_i=f(i)$.

As in the finite case, the probability function assigns to each $u \in \mathcal{U}$ a value $\mathrm{P}[u] \in[0,1]$. The basic requirement that the probabilities sum to one (equation (6.1)) is the requirement that the infinite series $\sum_{i=1}^{\infty} \mathrm{P}\left[u_i\right]$ converges to one. Luckily, the convergence properties of an infinite series whose terms are all non-negative is invariant under a re-ordering of terms (see §A4), so it does not matter how we enumerate the elements of $\mathcal{U}$.
Example 6.29. Suppose we flip a fair coin repeatedly until it comes up “heads,” and let the outcome $u$ of the experiment denote the number of coins flipped. We can model this experiment as a discrete probability distribution $\mathbf{D}=(\mathcal{U}, \mathrm{P})$, where $\mathcal{U}$ consists of the set of all positive integers, and where for $u \in \mathcal{U}$, we set $\mathrm{P}[u]=2^{-u}$. We can check that indeed $\sum_{u=1}^{\infty} 2^{-u}=1$, as required.

One may be tempted to model this experiment by setting up a probability distribution on the sample space of all infinite sequences of coin tosses; however, this sample space is not countably infinite, and so we cannot construct a discrete probability distribution on this space. While it is possible to extend the notion of a probability distribution to such spaces, this would take us too far afield.

## 数学代写|数论代写Number Theory代考|Random variables

$$\text { 让 } \mathbf{D}=(\mathcal{U}, \mathrm{P}) \text { 是一个概率分布。 }$$

## 数学代写|数论代写Number Theory代考|Expectation and variance

$$\mathrm{E}[X]:=\sum_{u \in \mathcal{U}} X(u) \cdot \mathrm{P}[u] .$$

$$\mathrm{E}[X]=\sum_{x \in \mathcal{X}} \sum_{u \in X^{-1}(x)} x \mathrm{P}[u]=\sum_{x \in \mathcal{X}} x \cdot \mathrm{P}[X=x] .$$

$$\mathrm{E}[f(X)]=\sum_{x \in \mathcal{X}} f(x) \mathrm{P}[X=x] .$$

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