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Just as the title, it is 2.7 in Ross's book (Introduction to Probability Models).

Consider arbitrary events A1,...,An, and let X denote the number if these events that occur. We will determine the probability mass function of X. To begin, for 1<= k <= n, let

$S_k$ = $\sum_{i_1<...<i_k} P(A_{i1}...A_{ik})$

....

Finally, it proves P(X>=k) = $\sum_{j=k}^n (-1)^{k+j} {j-1\choose {k-1}}S_j$

I had no idea what does "The distribution of the Number of Events that Occur" means?

Edit:

a) What does the expansion of $S_k$ looks like? like $\sum_i P(E_i E_j E_k E_l)$ in $P(E_1\bigcup E_2 \bigcup ... \bigcup E_n)$ = $\sum_i P(E_i)$ - $\sum_i P(E_i E_j)$ + $\sum_i P(E_i E_j E_k)$ - $\sum_i P(E_i E_j E_k E_l)$ + ... + $(-1)^{n+1} P(E_i E_j ... E_n) $

b) Does it has a countable sample space?

Please give me an example about such a topic, any reference are welcome.

Thanks in advance

Related question:

The Probability $P_{[m]}$ that exactly $m$ among the $N$ events $A_1,\dots,A_N$ occur simultaneously

evergreenhomeland
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  • It means the distribution of the random variable $X$ which is defined as the number of events $A_i$ which occur. For example, suppose $n=2.$ If neither $A_1$ nor $A_2$ occurs, then $X=0;$ if $A_1$ and $A_2$ both occur, then $X=2;$ if $A_1$ occurs but $A_2$ does not occur, or if $A_2$ occurs but $A_1$ does not occur, then $X=1.$ – bof Jan 10 '18 at 11:25
  • @bof In case of your example, what is the range of X? [0,2]? – evergreenhomeland Jan 10 '18 at 16:19
  • The number of occurrences is a discrete variable, its range is ${0,1,\dots,n}.$ If $n=3$ the range of $X$ is ${0,1,2}.$ – bof Jan 10 '18 at 18:54
  • You should check out the Bonferroni inequalities, and the Inclusion-Exclusion Principle. Specifically, read the special case section of the second link, noting that because we are looking at $P[X\geq k]$ we don't care which events occur, only that there are at least $k$ of them, and thus this gives us the binomial coefficients through the special case. – Ryan Warnick Jan 11 '18 at 01:48
  • @bof Thanks. So X represents types of distinct events that occurred? – evergreenhomeland Jan 11 '18 at 02:49
  • @RyanWarnick Thank you very much. – evergreenhomeland Jan 11 '18 at 03:10
  • Which chapter? I don't see the materials you mentioned in 2.7. –  Jan 16 '18 at 20:11
  • @Jack My text is 11th edition. it is section 2.7. The previous section 2.6 is Moment Generating Functions, and the next section 2.8 is Limit Theorems. – evergreenhomeland Jan 18 '18 at 03:16

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