# [Probability](https://math.stackexchange.com/questions/tagged/faq+probability) [tag:probability]

 - "Two Children Puzzle / Boy Born on a Tuesday" and variants: 
   - https://math.stackexchange.com/questions/15055
   - https://math.stackexchange.com/questions/33566
   - https://math.stackexchange.com/questions/4400

 - A "liar paradox" variant: https://math.stackexchange.com/questions/76491/
 - https://math.stackexchange.com/questions/132722
 - Expectation as tail probability (complementary CDF) integral/sum.  
    - Continuous (generalized to any $E[X^p]$ in the answer): [Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$](https://math.stackexchange.com/questions/172841).

    - Discrete:  [Find the Mean for Non-Negative Integer-Valued Random Variable](https://math.stackexchange.com/questions/843845)

    - Measure-theoretic treatment: [Rigorous proof that $\int_{\Omega}X\;dP=\int_{-\infty}^{\infty}xf(x)\;dx$](https://math.stackexchange.com/questions/402640)

 - Probability for roots of a quadratic equation to be real, where the coefficients are iid random variables with distribution:
  - [discrete uniform$\{1,2, 3,4,5,6\}$ from standard die](https://math.stackexchange.com/questions/120083), whereas [322939](https://math.stackexchange.com/questions/322939) is also a good candidate.
<sup>There are isolated posts regarding discrete uniform of different ranges, and continuous uniform, exponential distribution, etc.</sup>

 - Coupon collector's problem [tag:coupon-collector]
   - standard version
     - expectation: https://math.stackexchange.com/questions/28905 (duplicated at https://math.stackexchange.com/questions/782769/)
     - probability distribution: https://math.stackexchange.com/questions/379525/
   - only desired items
     - expectation: https://math.stackexchange.com/questions/2912436 (duplicated at https://math.stackexchange.com/questions/3682695)
     - probability distribution: https://math.stackexchange.com/questions/909313 (duplicated at https://math.stackexchange.com/questions/2488656)
   - batches / groups / packets
     - fixed number per packet, distinct coupons:
         - expectation: https://math.stackexchange.com/questions/1375470 (duplicated at https://math.stackexchange.com/questions/2147576 and https://math.stackexchange.com/questions/3840627)
         - probability distribution: https://math.stackexchange.com/questions/3396686
     - fixed number per packet, repetitions allowed:
         - expectation: https://math.stackexchange.com/questions/131664 (duplicated at https://math.stackexchange.com/questions/1600817)
         - probability distribution: https://math.stackexchange.com/questions/1595993
     - random number per packet, distinct coupons: https://math.stackexchange.com/questions/1729848/
    - unequal probabilities:
      - mean and variance: https://math.stackexchange.com/questions/3453965/
      - probability distribution: https://math.stackexchange.com/questions/4410002

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# [Statistics](https://math.stackexchange.com/questions/tagged/faq+statistics) [tag:statistics]

 - Upper tail inequality for the standard normal distribution: 
    - https://math.stackexchange.com/questions/28751,  
    - https://math.stackexchange.com/questions/74151/

 - Normal approximation (CLT) for discrete Binomial: ... (to be continued... okay, this task is more complicated than expected)