0
votes
1answer
26 views

Is there a name for sum over one set divided by the cardinality of another set?

What is the summation of one set real numbers divided by the cardinality of another set called? $$A \subset\mathbb R$$ $$\frac{\sum A}{|B|}$$ I will try and be specific to my problem because I lack ...
2
votes
3answers
72 views

Expectations and variance with rolling a dice 10 times

Let's say you roll a fair dice 10 times and X is the number of sides that never show up. (i.e. Roll 1 - 10 = 1424145221, X = 2 because 3 and 6 never show up) Values of $N=0,1,2,3,4,5.\\ P(N=6) = 0$ ...
0
votes
1answer
17 views

Show that the normal equations are identical to $\frac{\partial}{\partial\theta_j}QS(\theta)=0~\forall~j=1,\ldots,k$

Let the quadratic sum be given by $QS(\theta)=\sum_{i=1}^{n}(y_i-x^i\theta)^2$, with $y=(y_1,\ldots,y_n)^T, \theta=(\theta_1,\ldots,\theta_k)^T$ and $$ X=\begin{pmatrix}x_{11} & \ldots ...
0
votes
0answers
11 views

Normalizing the Second Moment of $n$ Discs.

Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($\|P_i - P_j\|\geq d, i\neq j$). Graham and Sloane use the second moment as a measure of compactness for ...
2
votes
1answer
204 views

Derivative of double summation and dot notation?

I am trying to differentiate the following summation: $$ L(\mu, \tau_1, \ldots, \tau_i)= \sum_{i=1}^v \sum_{t=1}^{r_i} (y_{it}-\mu - \tau_i)^2 $$ $$\frac{dL}{d\mu} = y_{\cdot\cdot}-n\mu - ...
0
votes
2answers
62 views

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$.

Show that $\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^{n} (x_i - \bar{x})(y_i)$. I fell into this hole where I keep finding that $\sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i ...
0
votes
1answer
73 views

Should be simple inductive proof

Establish the following recursion relations for means and variances. Let $\overline{X}_n$ and $S_n^2$ be the mean and variance, respectively, of $X_1,\dots,X_n$. Then suppose another observation, ...
0
votes
2answers
92 views

Need help with expectation of summation for power of Gaussian variable.

I am trying to derive a formula, and getting stuck on a part of the derivation. Basically what I have is the following: $$ P = \sum_{n=0}^{N-1} \sum_{k=0}^{N-1} \mathbb{E} \Big[x^2[n] x^2[k] \Big] $$ ...
0
votes
1answer
28 views

How can i simplify the following term to get the right side?

$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$ where, $n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ ...
0
votes
1answer
64 views

Solving simultaneous equations for Weibull distribution parameter estimation

What is the best way to solve the equations below using Matlab. I have the values for i, Ni, k, T and am in need of values for rho, beta and alpha. i goes up to 12 Thank you
0
votes
2answers
106 views

Sum in Probability $\sum_{n=k}^\infty \frac{(\lambda\cdot s)^n}{n!}e^{-\lambda s} \binom{n}{n-k}p^{n-k}(1-p)^k$

I am trying to evaluate a sum consisting of a possion distribution, multiplied by a binomial distribution. I already know by the power of maple that $$ \sum_{n=k}^\infty \frac{(\lambda\cdot ...
2
votes
1answer
50 views

How $\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!}=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$

$$P(X\geq m)=\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!};m=0,1,...$$ Show that for any $m=1,2,...$ $$P(X\geq m)=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$ I couldn't derive it also ...
1
vote
2answers
151 views

Can I get a little help proving equality between a summation and integral?

Prove$$\sum_{k=0}^x \binom{n}{k}p^{k}(1-p)^{n-k} =(n-x)\binom{n}{x}\int_{0}^{1-p}t^{n-x-1}(1-t)^{x}dt.$$ Can someone show me the steps please? Here is the hint my book gave me: "Integrate by parts ...
0
votes
0answers
24 views

How $\sum_{h=1}^{L}\sum_{h=1}^{L}W_h^2S_h^2=(\sum_{h=1}^{L}W_hS_h)^2$?

Suppose a population is divided into $L$ strata. $N_h=$Total number of units in the $h^{th}$ stratum $N=\sum_{h=1}^{L}N_h$ $W_h=$stratum weight and $W_h=\frac{N_h}{N}$ $S_h^2=$stratum population ...
1
vote
1answer
64 views

Why does this interchanging of derivative and sum work?

I'm reading a stats book and, for a geometric distrubution ($E[Y]=p \sum_{y=1}^{\infty}yq^{y-1})$ it makes the claim that since $\displaystyle \frac{d}{dq}(q^y)=yq^{y-1}$ hence $\displaystyle ...
1
vote
1answer
169 views

Can someone explain the intuition behind this moment generating function identity?

If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= ...
2
votes
2answers
213 views

Double Summation: Need help to handle $ i \neq j $ : $ \sum_{i=0 \to 7,\ j=1 \to 8,\ i\neq j} (8i + j) $

[Q1]. Can I ? ( write the same summation as ) : $$ \sum_{i=0, i \neq j}^7 \sum_{j=1}^8 (8i + j) \tag{1}$$ I tried to solve the following Summation as follows: Let i = m-1 then, $ \sum_{i=0,\ i ...
0
votes
2answers
27 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 $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
3
votes
4answers
300 views

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$

How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ i have tried to do that by the following procedure: $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$ ...
-1
votes
1answer
362 views

Expected value of sum squared is sum of expected value squared?

Consider the following expression: $ X \sim BIN(1, p) $ $ Var(\bar{X})=Var(\frac{\sum_i{X_i}}{n}) = \frac{1}{n^2} Var(\sum_i X_i) = \frac{1}{n^2} \left( \sum_i E(X_i^2) - ( \sum_i E(X_i) )^2 \right) ...
2
votes
1answer
33 views

Problem with a summation suffix.

Please can someone tell me why we drop the summation of i here? $S = \sum_{i,j}(y_{ij}-\mu -\alpha_i)^2$ $\frac{dS}{d\alpha_i} = \sum_{j}(y_{ij}-\mu -\alpha_i)$ It's part of a question which is ...
0
votes
1answer
121 views

Skewness of a sum with a positive summand

Let $X$ and $Z$ be two random variables with finite third moment, and let $Z>0$. Is it true that the skewness of $X+Z$ is greater or equal than that of $X$? Such a relation clearly holds for the ...
-2
votes
4answers
1k views

Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial

The problem: X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$ Expected value is $\sum{x_ip(x_i)}$ But ...
2
votes
2answers
417 views

Geometric Distribution $P(X\ge Y)$

I need to show that if $X$ and $Y$ are idd and geometrically distributed that the $P(X\ge Y)$ is $1\over{2-p}$. the joint pmf is $f_{xy}(xy)=p^2(1-p)^{x+y}$, and I think the only way to do this is to ...
1
vote
2answers
174 views

How to break apart this sum?

I have a summation I need to break apart but I can't figure it out http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape10/PQDD_0027/MQ50799.pdf $p.15$, right after line $(3.8)$ Going from the ...
1
vote
0answers
5k views

Finding the Moment Generating function of a Binomial Distribution

Suppose $X$ has a $Binomial(n,p)$ distribution. Then its moment generating function is $$ M(t) = \sum_{x=0}^x {n \choose x}p^x(1-p)^{n-x} \\ =\sum_{x=0}^{n} {n \choose x}(pe^t)^x(1-p)^{n-x}\\ ...
1
vote
2answers
143 views

Linear vs Logarithmic weighted sum?

So I have some voting data. There were 25 possible things to vote for, and each voter had to fill out a 10 position ballot, ie each user picked 10 from the 25 possible things for each ballot. ...
2
votes
0answers
295 views

sum with permutations

Let $a$ be vector in $R^{2m}$. And let $S_{2m}$ be group of all permutations on the set $\{1,\dots,2m\}$. I would like to calculate $$ \sup_{\pi\in S_{2m}}\sum_{d(\sigma, ...