Intereting Posts

Intuition for the Product of Vector and Matrices: $x^TAx $
Prove $(2, x)$ is not a free $R$-module.
A basic question on expectation of distribution composed random variables
Quadratic Reciprocity: Determine if 11 is a quadratic residue $\mod p $for primes of the form: 44k+5?
Is there a bijection between $\mathbb{Q}$ and $\mathbb{Q}_{>0}$?
No group of order 36 is simple
How many surjective functions are there from $A=${$1,2,3,4,5$} to $B=${$1,2,3$}?
Is there a number whose absolute value is negative?
Definition of a Cartesian coordinate system
The sum of the first $n$ squares is a square: a system of two Pell-type-equations
Distributive Law and how it works
Number of onto functions
Is there a common symbol for concatenating two (finite) sequences?
About evaluating $\mathcal{L}^{-1}_{s\to x}\bigl\{\frac{F(s)}{s}\bigr\}$ by considering contour integration with different entire functions $F(s)$
Sum of all natural numbers is 0?

I’d like to know if some results are known on the following type of random variables: for parameter $p\in[0,1]$ (for my purposes, $p < \frac{1}{2}$, and even $p \ll 1$) and $n \geq 1$, we let $X$ be a random variable following a Binomial$(n,p)$ distribution, and define $$Y \stackrel{\rm def}{=} \min(X, n-X).$$

Then, as a function of $p$ and $n$: what is the distribution of $Y$? In particular, what is its expectation $\mathbb{E}[Y]$, and what are the best concentration bounds one can obtain around $\mathbb{E}[Y]$? (Does anything like Hoeffding/Chernoff hold?)

As a small remark: for the range of $p$ I am interested in, we should have $\mathbb{E}[Y] \approx \mathbb{E}[X] = pn$, since the probability that $X > \frac{n}{2}$ is exponentially small (in $n$). Big discrepancies between $X$ and $Y$ should only appear when $pn\in [n/2-O(\sqrt{n}), n]$, and for $p$ outside this range $X$ and $Y$ should have distributions *very* close (in total variation distance). But is there (a) an exact expression for $\mathbb{E}[Y]$, or one that holds even for moderately small $n$? and (b) good concentration bounds for $Y$?

- How to prove Poisson Distribution is the approximation of Binomial Distribution?
- The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?
- Finding mode in Binomial distribution
- Sum of two independent binomial variables
- Why is the sum of the rolls of two dices a Binomial Distribution? What is defined as a success in this experiment?
- Conditions needed to approximate a Binomial distribution using a Normal distribution?

For instance, for $n=2$ we get the exact value $\mathbb{E}[Y] = 2p(1-p) = 2p + O(p^2)$, and for $n=3$ we obtain $\mathbb{E}[Y] = 4p + O(p^3)$.

- Expectation of an exponential function
- How can I solve bins-and-balls problems?
- Average Distance Between Random Points on a Line
- Expected Value Of Dice Rolling Game
- Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$
- Distance of a test point from the center of an ellipsoid
- Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
- Chance of 7 of a kind with 10 dice
- product distribution of two uniform distribution, what about 3 or more
- Why not defining random variables as equivalence classes?

- Prove there is no contraction mapping from compact metric space onto itself
- Is there always a prime between a prime and prime plus the index of that prime?
- Modus Operandi. Formulae for Maximum and Minimum of two numbers with a + b and $|a – b|$
- A combinatorial proof of Euler's Criterion? $(\tfrac{a}{p})\equiv a^{\frac{p-1}{2}} \text{ mod p}$
- Integration of a function with respect to another function.
- Modular equation for a quotient of eta functions
- Is this proof for Theorem 16.4 Munkers Topology correct?
- If $a^2+b^2=1$ where $a,b>0$ then find the minimum value of $a+b+{1\over{ab}}$
- Eigenvalues of product of a matrix and a diagonal matrix
- How would one go about proving that the rationals are not the countable intersection of open sets?
- Derivative of the trace of matrix product $(X^TX)^p$
- $A \in {M_n}$ is normal.why the range of $A$ and ${A^*}$ are the same?.
- Hatcher Exercise 2.2.38
- A 3rd grade math problem: fill in blanks with numbers to obtain a valid equation
- Recurrence relation (linear, second-order, constant coefficients)