Intereting Posts

Principle of Transfinite Induction
The 'sine and cosine theorem' – formulas for the sum and difference
Thurston's 37th way of thinking about the derivative
Use induction to prove that that $8^{n} | (4n)!$ for all positive integers $n$
Are there always at least $3$ integers $x$ where $an < x \le an+n$ and $\gcd(x,\frac{n}{4}\#)=1$
Why are x and y such common variables in today's equations? How did their use originate?
Constructing a NFA for the following language
Rings with a given number of (prime, maximal) ideals
The resemblance between Mordell's theorem and Dirichlet's unit theorem
Inner product on $C(\mathbb R)$
Nonlinear simultaneous equations
What is this pattern called?
Intriguing polynomials coming from a combinatorial physics problem
Measure on topological spaces
The free monoid functor is fully faithful?

Suppose that $X_1, \ldots, X_n$ follows Bernoulli distribution $B(1,p)$,

then what is the UMVUE of $p^s$ and $p^s + (1-p)^{n-s}$?

I suppose I should use the Lehmann–Scheffé theorem. Now $\overline{X}$ is a sufficient and complete statistic, I need to find a function of $\overline{X}$ whose expectation is $p^s$ and $p^s + (1-p)^{n-1}$. But I don’t know how to find such a function.

- Why do we subtract
- Is $g(u)= \frac{E }{E }$ decreasing in $u$
- “Empirical” entropy.
- Uniform distribution on the surface of unit sphere
- Monte-Carlo simulation with sampling from uniform distribution
- Something Isn't Right With My Parking

Any hint would be welcome!

- expected value of a function
- Truncated Mean Squared
- Is order of variables important in probability chain rule
- Probability that A meets B in a specific time frame
- Conditional and Total Variance
- Reviewing for exam: Chernoff bounds for a series of random variables
- Probability for “drawing balls from urn”
- Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$
- Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.
- Urn of balls: same marginal probability of red for all picks?

You have $$\operatorname{E}(X_1\cdots X_s) = p^s$$ if $s$ is an integer and $1\le s\le n,$ and if $1\le n-s\le n$, then $$\operatorname{E}(X_1\cdots X_s + (1-X_{s+1})\cdots(1-X_n)) = p^s + (1-p)^{n-s}.$$

The estimators that you need are the conditional expected values

$$

\operatorname{E}(X_1\cdots X_s\mid \overline X_n) \text{ and } \operatorname{E}((1-X_{s+1})\cdots(1-X_n)\mid \overline{X}_n).

$$

Because $\overline X_n$ is sufficient, the conditional expected values above do not depend on $p$ and are therefore observable and can be used as estimators.

\begin{align}

& \operatorname{E}\left(X_1\cdots X_s \mid \overline X_n = \frac x n\right) \\

= {} & \Pr\left( X_1\cdots X_s = 1 \mid X_1+\cdots+X_n = x \right) = \frac{\dbinom {n-s}{n-x}}{\dbinom n x}

\end{align}

So the Lehmann–Scheffé theorem says the UMVUE of $p^s$ is

$$

\frac{\dbinom{n-s}{n-(X_1+\cdots+X_n)}}{\dbinom n {X_1+\cdots+X_n}}.

$$

- The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
- On the number of Sylow subgroups in Symmetric Group
- Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ cannot be closed under scalar multiplication by $a \ne 0,1$
- Are the axioms for abelian group theory independent?
- What are some examples of classes that are not sets?
- Is there exist a homemoorphism between either pair of $(0,1),(0,1],$
- How to interpret Fourier Transform result?
- $f'(x) = g(f(x)) $ where $g: \mathbb{R} \rightarrow \mathbb{R}$ is smooth. Show $f$ is smooth.
- Combinatorics Distribution – Number of integer solutions Concept Explanation
- integration of function equals zero
- A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$
- Finding the circles passing through two points and touching a circle
- Will Division by Zero be Defined Eventually?
- How to express $\sin \sqrt{a-ib} \sin \sqrt{a+ib}$ without imaginary unit?
- Why do proof authors use natural language sentences to write proofs?