All Questions
26
questions
1
vote
1
answer
61
views
Find the expected value of $|H-T|$
Bob repeatedly throws a (fair) coin. For each throw, there is a $4\%$ chance that Bob decides to stop throwing the coin. He records the number of heads H and the number of tails T before he stops ...
- 2,031
1
vote
1
answer
153
views
When can we swap two sigmas in two infinite sums?
I have a space of elementary outcomes $\Omega = \{w_1, w_2, w_3, ... \}$
We have assigned a non-negative number $p(w)$ to every elementary outcome $w \in \Omega$ in such a way that
$\sum_{k=1}^\infty ...
- 12.7k
0
votes
1
answer
40
views
How to read and interpret the following mathematical notation? $\Gamma_{su} = \frac{P_sG_{su}}{\sum_{i\in S\{\backslash s\}}P_iG_{iu}+P_{AWGN}}$
I came across the following expression in a research paper:
$$\Gamma_{su} = \frac{P_sG_{su}}{\sum_{i\in S\{\backslash s\}}P_iG_{iu}+P_{AWGN}}$$
My query is how to read that small $s$ in denominator ...
- 71
1
vote
0
answers
41
views
Solving double sum for the PMF of random variable Z=X+Y-1
Let $X,Y$ be two random variables with joint probability mass function
$$p_{X,Y}(k,l)=\begin{cases}
\frac{6}{\pi^2(k+l-1)^3},\, k,l\in\mathbb{N},\\
0, \text{ otherwise}.
\end{cases}$$
Now I need to ...
- 329
0
votes
2
answers
68
views
Calculating: $\sum_{s=0}^\infty\sum_{t=0}^s e^{-3}st\frac{1}{t!}\frac{2^{s-t}}{(s-t)!}$
After a considerable time of trying to calculate the following:
$$\sum_{s=0}^\infty\sum_{t=0}^s e^{-3}st\frac{1}{t!}\frac{2^{s-t}}{(s-t)!}$$
Assuming 0≤t≤s.I succeeded to reach an answer by using the ...
- 166
3
votes
1
answer
54
views
Calculating a sum $\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$
I want to calculate this sum, while $0<p<1$:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}$$
Is this correct:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}=\frac{1}{1-p}\cdot \frac{1-\frac{1}{(1-p)^k}}{1-\frac{1}{1-...
- 627
1
vote
1
answer
44
views
Calculating a sum sigma, with minimum $\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$
How do I calculate this sum:
$$\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$$
It is like a geometric sum, but it has the minimum which I do not know how to deal with.
I got this sum while calculating two ...
- 627
2
votes
1
answer
77
views
equality involving sums
Let $n\in\mathbb{Z}^+.$ Prove that for $a_{i,j}\in\mathbb{R}$ for $i,j = 1,\cdots, n,$
$$\left(\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\right)^2 + n^2\sum_{i=1}^n\sum_{j=1}^n a_{i,j}^2 - n\sum_{i=1}^n \left(\...
- 1,225
0
votes
1
answer
52
views
How did the author simplify this sum?
I have been looking at this derivation
The mean photon number is given by:
\begin{align}
\bar n&=\sum_{n=0}^\infty n\ \cal P_\omega(n)\\
&=\sum_{n=0}^\infty nx^n(1-x)\\
&=(1-x)x\ \frac{\...
- 13
0
votes
1
answer
72
views
some identity of summation and generalization
I see this way and idea from Simply Beautiful Art
profile here
Let : $S=\displaystyle\sum_{1≤k≤m≤n≤\infty}f(m)f(k)f(n)$
then :
$6S=\displaystyle\sum_{n,m,k≥1}f(m)f(k)f(n)+3\displaystyle\sum_{n,m≥...
- 2,323
0
votes
2
answers
62
views
Is inline summation index has different functionality, such as $\sum_{x,y} f(X,Y)$?
I find different papers use summation with a different style than other summation in the same paper. This thing repeats in more than one paper. Therefore I believe this is summation do different work ...
- 663
1
vote
2
answers
487
views
Finding the limit as $k$ tends to infinity of this sum
$\sum_{i=0}^{\lceil zk \rceil}{k\choose i} p^i(1-p)^{k-i}$
$z, p \in [0,1]$
I am looking to find the limit as $k$ tends to infinity but don't know how I would do this
- 125
2
votes
1
answer
189
views
Fundamental theorem of calculus - range instead of point by point
I read the following in a math article about continuous sample spaces:
We need to have P(Ω) = 1, i.e., P([0, T]) = 1. On the other hand, in the first experiment, all points in the interval [0, T] ...
- 277
1
vote
1
answer
208
views
Prove that H(X|Y)≥0
I am trying to prove the conditional entropy of X given Y is greater than or equal to 0.
I am told that the entropy $H(X)$ (according to Boltzmann's H) is equal to $$H(X)=\sum_{i=1}^n -P_i\log_2P_i$$
...
- 111
0
votes
1
answer
37
views
Proving expected size $\ge$ average
With groups of sizes $x_1, x_2, \dots, x_n$ adding to $G$, the average size is _____. The chance of an individual belonging to group 1 is _____. The expected size of his or her group is $E(x) = x_1 (...
- 1,345