All Questions
26
questions
4
votes
6
answers
503
views
Proof of equation with binomial coefficients: $\sum\limits_{k=1}^{n} (k+1) \binom{n}{k} = 2^{n-1} \cdot (n+2)-1$ [duplicate]
$$\sum\limits_{k=1}^{n} (k+1) \binom{n}{k} = 2^{n-1} \cdot (n+2)-1$$
Maybe it's simple to prove this equation but I'm not sure how to get along with the induction. Any hints for this? Or may I use ...
- 824
3
votes
1
answer
55
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:
$$\sum_{i=1}^{k-1}\frac{1}{(1-p)^i}=\frac{1}{1-p}\cdot \frac{1-\frac{1}{(1-p)^k}}{1-\frac{1}{1-...
- 627
3
votes
1
answer
88
views
Any simpler form for $ \frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$
Is there any simpler form for the following expression:
$$
\frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$
Because I have to compute this ...
- 1,445
3
votes
2
answers
275
views
$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}= ?$
Could you please help me. How do I sum the following:
$$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}$$
If the summation had started at 0, then it would be simply an ...
- 31
3
votes
1
answer
628
views
Is this infinite sum always less than zero?(+500pts bounty for the correct answer)
I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum:
$$\frac{\partial}{\partial d}\sum_{n=1}^\infty n \...
- 7,892
2
votes
1
answer
77
views
equality involving sums
Let $n\in\mathbb{Z}^+.$ Prove that for $a_{i,j}\in\mathbb{R}$ for $i,j = 1,\cdots, n,$
$$\left(\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\right)^2 + n^2\sum_{i=1}^n\sum_{j=1}^n a_{i,j}^2 - n\sum_{i=1}^n \left(\...
- 1,225
2
votes
1
answer
189
views
Fundamental theorem of calculus - range instead of point by point
I read the following in a math article about continuous sample spaces:
We need to have P(Ω) = 1, i.e., P([0, T]) = 1. On the other hand, in the first experiment, all points in the interval [0, T] ...
- 277
1
vote
4
answers
3k
views
How to calculate this double sum? [closed]
This occurred in a probability problem where I have to calculate the invariant $c$ which equals to $N$ divided by the following double summation:
$$\sum_{n=0}^{N} \sum_{k=0}^N |k-n|$$
1
vote
1
answer
311
views
Expected value of a Poisson sum of confluent hypergeometric functions
How to compute the expected value of a Poisson sum of the following confluent hypergeometric function:
$$
\sum_{y=1}^{Y} {}_1F_1(y,1,z)
$$
where y is positive integer taking values from the Poisson ...
- 510
1
vote
1
answer
153
views
When can we swap two sigmas in two infinite sums?
I have a space of elementary outcomes $\Omega = \{w_1, w_2, w_3, ... \}$
We have assigned a non-negative number $p(w)$ to every elementary outcome $w \in \Omega$ in such a way that
$\sum_{k=1}^\infty ...
- 12.6k
1
vote
2
answers
487
views
Finding the limit as $k$ tends to infinity of this sum
$\sum_{i=0}^{\lceil zk \rceil}{k\choose i} p^i(1-p)^{k-i}$
$z, p \in [0,1]$
I am looking to find the limit as $k$ tends to infinity but don't know how I would do this
- 125
1
vote
1
answer
44
views
Calculating a sum sigma, with minimum $\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$
How do I calculate this sum:
$$\sum_{i=1}^{\min(n,k-1)}\frac{1}{(1-p)^i}$$
It is like a geometric sum, but it has the minimum which I do not know how to deal with.
I got this sum while calculating two ...
- 627
1
vote
1
answer
61
views
Find the expected value of $|H-T|$
Bob repeatedly throws a (fair) coin. For each throw, there is a $4\%$ chance that Bob decides to stop throwing the coin. He records the number of heads H and the number of tails T before he stops ...
- 2,031
1
vote
0
answers
41
views
Solving double sum for the PMF of random variable Z=X+Y-1
Let $X,Y$ be two random variables with joint probability mass function
$$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}$$
Now I need to ...
- 329
1
vote
1
answer
210
views
Prove that H(X|Y)≥0
I am trying to prove the conditional entropy of X given Y is greater than or equal to 0.
I am told that the entropy $H(X)$ (according to Boltzmann's H) is equal to $$H(X)=\sum_{i=1}^n -P_i\log_2P_i$$
...
- 111