All Questions
10
questions
2
votes
1
answer
47
views
Double Sum to Product Derivation
The function after the double-sigma sign can be separated into the
product of two terms, the first of which does not depend on $s$ and
the second of which does not depend on $r$. Source
Is the ...
- 1,834
1
vote
0
answers
47
views
Multiple Sigma Notation and Expected Value
I've been struggling with expected value calculations that involve multiple or nested sigma notations. For instance, the solution to a problem I was working on is:
$X=\Sigma_{i=1}^{10}X_i\implies E[X]...
- 111
4
votes
3
answers
763
views
Probability you end dice rolling sequence with 1-2-3 and odd total number of rolls
Here's a question from the AIME competition:
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an ...
- 1,415
-1
votes
3
answers
41
views
How does $\sum_{k=m+1}^{\infty}(1-p)^{k-1}p$ become $(1-p)^m p \sum_{k=0}^\infty (1-p)^k$?
What sum rules are applied from line 1 to 2 below? I get the constant rule in regard to $p$, but that's about it.
$$\begin{align}
P(X > m) &= \sum_{k=m+1}^{\infty}(1-p)^{k-1}p \tag{1}\\
&= (...
- 1
0
votes
1
answer
40
views
Two sums out of a product
Problem
Assuming $\theta \in [0, 1], y_i \in \{0, 1\}$,
I'm having trouble deriving the expression on the right from the one on the left:
$$
\begin{align}
\prod_i \theta^{y_i}(1-\theta)^{(1-y_i)} ~~~...
- 2,055
0
votes
1
answer
31
views
Likelihood ration Algebraic issue
I have got the following likelihood:
$$l(p) = C + xlog(p) + (n-x)log(1-p)$$
I have got that $\theta_0 = 1/3$ and $\theta_1 = 1/2$
All I need to do is find the correct value of the log likelihood ...
- 149
1
vote
4
answers
94
views
7
votes
2
answers
195
views
Computing the sum $\frac{1}{(\frac{1}{n}-p)^{2}}+\frac{1}{(\frac{2}{n}-p)^{2}}+\ldots+\frac{1}{(1-p)^{2}}$
Let $p\in(0,1)$ and $n$ be a finite positive integer. How to compute the following sum
\begin{equation}
\frac{1}{(\frac{1}{n}-p)^{2}}+\frac{1}{(\frac{2}{n}-p)^{2}}+\ldots+\frac{1}{(\frac{n-1}{n}-p)^{2}...
- 638
0
votes
2
answers
32
views
A small calculation .
How
$\sum_{k=0}^n (-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^k$
i got it
$\sum_{k=0}^n(-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^{2n-k}$
And
is that $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
- 1,457
3
votes
2
answers
124
views
Finding expected value of a binomial
I'm trying to solve the following question: You and n other people (so n+1 people) each toss a probability-p coin, with $0\le P \le 1$. Then each person who got a head will split some arbitrary amount ...
- 33