Intereting Posts

Materials for self-study (problems and answers)
Proving that $\sqrt{2}+\sqrt{3}$ is irrational
Pseudo-Cauchy sequence
Must a weakly or weak-* convergent net be eventually bounded?
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) – \arctan(y)| $ is an incomplete metric space
Factoring inequalities on Double Summation (Donald Knuth's Concrete Mathematics)
A new combinatorics identity— similar to Catalan number
Five Cubes in Dodecahedron
Examples of a non commutative division ring
How many arrangements of the letters in the word CALIFORNIA have no consecutive letter the same?
The Galois Extension of $\mathbb Q$ with cyclic group of prime order as its Galois group
Why are $e$ and $\pi$ so common as results of seemingly unrelated fields?
Reference request: Vector bundles and line bundles etc.
$\gcd(b^x – 1, b^y – 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$
Example I.4.9.1 in Hartshorne (blowing-up)

In the coupon collectors problem, there are $n$ unique coupons and each cereal box has 1 coupon only. I would like to modify the problem such that there are $m$ boxes of cereal in total and each box has $c_i (1 \le c_i \le n)$ number of coupons.

Then how many boxes of cereal do I need to buy to have $n$ unique coupons?

- Characteristic function of product of normal random variables
- Book suggestion for probability theory
- Domino Probability Problem
- Everything in the Power Set is measurable?
- combining conditional probabilities
- Convergence of Random Variables in mean

- Expected value - continuous random variable
- Why does this not seem to be random?
- Why is stopping time defined as a random variable?
- If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.
- Roulette betting system probability
- What are the restrictions on the covariance matrix of a nonnegative multivariate distribution.
- Choosing two random numbers in $(0,1)$ what is the probability that sum of them is more than $1$?
- Probability of selecting different kinds of items from a set of items
- Is conditional expectation with respect to two sigma algebra exchangeable?
- What is the difference between a probability distribution on events and random variables?

Since the coupons within a box are not guaranteed to be different, basically we are just buying boxes in batches of $c_i$.

As in the original coupon problem, we expect to have to buy $n \log n$ original boxes to make for a complete set of $n$ unique coupons. So if each new box contains $c$ coupons, we expect to need to buy $\dfrac{n \log n}c$.

However, without any specification on how the $c_i$ are distributed for the new boxes we cannot make an estimate of the number of boxes we need to buy in the new situation.

Let us tackle the general situation where each box has the same probability distribution, say chance $p_i$ to contain $i$ boxes, $1 \le i \le n$. Then each box is expected to contain $$E = \sum_{i=1}^n ip_i$$ coupons. It follows that we expect to have to buy $$\frac{n \log n}E$$ boxes to complete a set.

- Mathematical Analysis advice
- Understanding Primitive Polynomials in GF(2)?
- A representation of Dirac-$\delta$
- Proof of “continuity from above” and “continuity from below” from the axioms of probability
- How to prove continuity of $e^x$.
- What's a good book on advanced linear algebra?
- How is the derivative truly, literally the “best linear approximation” near a point?
- Proof that the Integral of a Positive Function is Positive?
- Power of a matrix, given its jordan form
- Showing $\sin(\bar{z})$ is not analytic at any point of $\mathbb{C}$
- Number we know all prime numbers less than
- Find the limiting value of $S=a^{\sqrt{1}}+a^{\sqrt{2}}+a^{\sqrt{3}}+a^{\sqrt{4}}+…$ for $0 \leq a < 1$
- Does the Pell-like equation $X^2-dY^2=k$ have a simple recursion like $X^2-dY^2=1$?
- Algorithm to compute Gamma function
- How do Taylor polynomials work to approximate functions?