Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [summations]

68 questions
49
votes
5 answers

Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a general set of Gamma random variables. I have…
OSE
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14
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1 answer

What does it mean if the median or average of sums is greater than sum of those of addends?

I'm analyzing the distribution of network latency. The median upload time (U) is 0.5s. The median download (D) time is 2s. However, the median total time (for each data point, T = U + D) is 4s. What conclusions could be drawn knowing that the median…
David Faux
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1 answer

Summation of Two random variable

Suppose $X$ is random variable with PDF $f(X)=2(x-1)$, $1 \le x \le 2$; $Y$ is a random variable with a triangle pdf with minimum at $2$, mode at $2.5$, and maximum at $3$. Is it possible to define a random variable like $Z$ which is the summation…
Katatonia
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2 answers

Sum of random variables without central limit theorem

I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution function of sum of independent random variables ?…
upol94
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multi stage binomial "process"

I wish to model the retransmission time of a file that divided into K blocks. I know the successful blocks of first transmission obey the binomial distribution $$ X_1 \sim \text B(K,p) $$ , p is the successful probability of packet delivery. Then,…
robit
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1 answer

Sum of products in an expected value

A box contains $n$ balls numbered from 1 to $n$. Suppose you take a ball at a time, putting it back on the box, until you pick a ball twice. How many balls are you expected to take from the box? Let $X$ be the r. v. of interest. Its support is every…
Luke
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What can we conclude about the distribution of the sum of two random variables?

If we know, for independent random variables $X$ and $Y$, $P(X>x)\leq0.05$, and $P(Y>y)\leq0.05$, can we say anything about $P(X+Y>x+y)$? Can we be certain that it is less than $0.05$? Under what conditions could we say that?
Masood_mj
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2 answers

Distribution of sum of function of two random variables

Let $\{x_1, \ldots, x_n\}$ be a set of $n$ i.i.d. samples from a distribution $p(x)$. I would like to evaluate the distribution of the sum $$ S = \sum_{1\leq i
Till Hoffmann
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1 answer

Using Poisson distribution to evaluate summations

I'm interested in how to use a Poisson distribution to evaluate $\sum\limits_{x=0}^\infty \frac{(x^2-x+1)(2^x)}{x!}$ I see that this is similar to the general pmf form of $\frac{2^{x}}{x!}$. My strategy is to simplify the summation such that it…
Tim
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4
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1 answer

Measuring share contribution of each var/cov term to the standard deviation of a sum of variables

Say, for a simple example, I have a random variable $X = \alpha_1 X_1 + \alpha_2 X_2$, where $X_i$ are random variables and $\alpha_i$ are weights. I then calculate the standard deviation of $X$ as $\sigma_X = \sqrt{(\alpha_1)^2 \mathrm{Var}(X_1) +…
XtremeCurling
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3
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1 answer

Is there a way to prove $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$ without resorting to summations?

I would like to show that $\mathbf{\hat{Y}}^T\mathbf{e}=\mathbf{0}$. I can solve this by saying that it is equivalent to showing $\sum e_i\hat{y}_i=0$. However, I'm wondering if there is a way to employ matrix methods to show that…
Jen Snow
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1 answer

Summation of combinations up to $r-1$ terms

I am trying to come up with a simplified expression for $$\sum_{k=r}^{n}\binom{n}{k}$$ Choosing $x=y=1$ in Binomial theorem, I have $$2^n = \sum_{k=0}^{n}\binom{n}{k}$$ $$2^n = \sum_{k=0}^{r-1}\binom{n}{k} + \sum_{k=r}^{n}\binom{n}{k}$$…
Emon
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2 answers

Summation of the Matrix of a Squared Sum

I find that: $$\Big(\sum_{i = 1}^n y_i\Big)^2 = \sum_{i=1}^m y_i^2 + \sum^m_{i\neq j}y_iy_j$$ where $m=(n^2-n)=n(n-1)$, the first part of the right side is the sum of the main diagonal of the matrix, and the second part is the sum of the upper and…
Archimede
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Deriving OLS estimates using method of moments

I've worked the slope all the way down to $\sum [x_i(y_i - \bar{y})] = \hat\beta_1 \sum[x_i(x_i - \bar{x})]$ But I can not figure out how to show the steps for: $\sum[x_i(y_i - \bar{y})] = \sum(x_i - \bar{x})(y_i - \bar{y})$ and $\hat \beta_1…
Travis
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Sum of function of normally distributed values

I have a given number, $v$, given by $$v = \sum_{i=1}^N\left\{v_i \cdot \left[m \cdot \left(1 - b_i \cdot \mathbb{1}_{\geq w}(v_i)\right) - b_i \cdot \mathbb{1}_{< w}(v_i) \right] - a \cdot \mathbb{1}_{\geq w}(v_i) \right\}$$ where all the $v_i$ are…
Pedro Carvalho
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