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This e-book is rather interesting: Introduction to Probability, by Grinstead & Snell, at Dartmouth. Have also a look at their "Chance" web site. If you are intereseted in statistical applications, those may help: NIST Engineering Statistics Handbook Statistics Textbook from StatSoft

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Mathematics for Computer Science by Eric Lehman and Tom Leighton This ebook has an excellent chapter on probability, it's not rigorous, but it builds very good intuition.

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Since you're working with a discrete random variable, the probability of X being lower than b equals the probability of X being equal to or lower than the number before b. Imagine we're working with a fair dice: $P(2\leq X < 5) = P(2\leq X \leq 4)$. Sounds quite fair, doesn't it? I think you can answer your question yourself now :)

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There are very many applications of stochastic processes. To name a few, gambling, statistical sampling, insurance, communication networks, stock option pricing.

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The CDF of a discrete random variable $X$ is continuous everywhere except at those discrete points $x_i$ for which $P\{X=x_i\} > 0$. At these points of discontinuity, the limit from the right exists as does the limit from the left, but these two limits have different values. Let $F_X(a^{-}) = \lim_{x\uparrow a} F_X(x)$ denote the limiting value of ...

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Hint Win corresponds to a multiple of $1.5$ and loss to $0.5$. As multiplication commutes, you end up with $(1.5\times0.5)^5=0.75^5$ times of what you start with.

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Preliminary comment: The result follows immediately from the meaning of binomial distribution. But the problem is asking for a computation. We deal with the induction step. Assume that the sum $Y_n=X_1+\cdots+X_n$ has binomial distribution. This means that for all suitable $i$, we have $$\Pr(Y_n=i)=\binom{n}{i}q^i(1-q)^{n-i}.$$ We want to prove that ...

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Note that the Haar measure on $S^1$ is just normalised Lebesgue measure, and this is the quintessential setting of Fourier analysis on compact Abelian groups. A million things have been written on the subject even without a probabilistic interpretation. The invariance of Haar measure w.r.t. the group action $\theta \rightarrow \theta + \phi$ is simply a ...

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