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The probability that an even number of trials succeed is exactly $$ \frac12\biggl( 1 + \prod_{i=1}^n (1-2p_i)\Biggr).$$ This is a standard elementary application of probability generating functions.

This 1965 paper of Naus contains formulas for the exact probabilities. Turning them into the asymptotic results you want might take a little effort but since Naus' paper is cited in at least 255 plac …

It is true. Let $\chi_Z(x,y)$ be the indicator function of the set $\lbrace (x,y) \,|\, f(x,y)\in Z\rbrace$ for some set $Z$. The condition that the events $f(A_1,B)\in Z$ and $f(A_2,B)\in Z$ are in …

Looking at the discrete nonnegative case, using your notation, $$\sum_{i,j} (p_i-q_i)(p_j-q_j)|i-j| = -2 \sum_{i=0}^\infty \left( \sum_{j=0}^i p_j - \sum_{j=0}^i q_j\right)^2 \le 0,$$ where there are …

There are quite a lot of approximations more accurate than the pure normal distribution, with a large literature difficult to sort out. You can see several in this paper (Adv. Appl. Prob., 21 (1989) 4 …

The distribution of $X$ is called the Poisson-Binomial distribution and searching on that phrase will find a large amount of literature on it.

Just expand $ln(X+\alpha)$ as a Taylor series about $X=np$ and then do the binomial sum term by term. Use tail bounds on the binomial distribution to show that the error terms are meaningful. Without …

The broken logic in the book can bite in many places. Here is a simple example. If we make a random graph with $n$ vertices and edge probability $3/n$, the expected number of hamiltonian cycles goes …

CLT (sufficiently powerful version such as Berry-Esseen inequality) says that $Pr(X\ge c)\to\frac12$, so any event that has tiny probability for $X$ also has tiny probability for $X_{\ge c}$. So $E(X …

Define $g(\gamma) = \sum_{j\in J} \gamma^j$. The condition $E(X^2)\sim A\mu^2$ as $\mu\to\infty$ seems to be equivalent to $$ \frac{g(\gamma) g''(\gamma)}{(g'(\gamma))^2} \to A $$ as $\gamma\to 1$ fr …

$E[X^{2d}]/(2d)!$ is the coefficient of $x^{2d}$ in the Taylor expansion of $\cosh(x)^n$, since $\cosh(x)$ is the exponential moment generating function of each $X_i$. I don't know if there is a clos …

The total variation distance between $B(n,p)$ and Poisson($np$) goes to 0 as $n\to\infty$ whenever $p\to 0$. See http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2086772 for e …

(Partial answer) In GF(2), row and column permutations preserve the determinant. The equivalence classes under those operations are non-isomorphic bipartite graphs with $n$ vertices on each side and $ …

Interestingly, the probability that $t$ occurs for a standard die $n=6$ as $t\to\infty$ is $\frac27$, with exponential convergence towards that value. This is a standard generating function problem. T …

Each point in $[0,1)$ has probability $p=\prod_{i=1}^\infty (1-2^{-i})$ of being missed by all the intervals, so the expected measure of what is not missed is $1-p$. This is approximately 0.7112119; …