Questions tagged [sum]

The sum of two or more random variables.

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37 views

How is a convex combination of Dirichlet-distributed variables distributed?

Let $X = (X_1, \dots, X_K) \sim \operatorname{Dir}(\alpha_1, \dots, \alpha_K)$ and define the convex combination $Y = \sum_{i=1}^{K} c_i X_i$. In the case of $K=2$, the constraint $\sum_{i=1}^{K} X_i =...
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29 views

How to apply Lyapunov CLT to data

I have a situation where I have around 30 classes of variables with different means and variances (though the means aren't too far from eachother; think 4-7) and that the distributions are right ...
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1answer
183 views

2 approaches for Monte-Carlo : weighted sum of $\chi^2$ distribution and Moschopoulos distribution with Gamma distribution

If I take as definition of $a_{lm}$ following a normal distribution with mean equal to zero and $C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})$, and if I have a sum of $\chi^2$, can I write the 2 ...
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50 views

Probability that any element of a random unit-length vector is large [closed]

Given a vector $X \in R^n = \{x_1, x_2, ..., x_n\}$ drawn uniformly such that: $x_i \in [0, 1]$ for all $i$; and $\sum x_i = 1$, how would you find the probability that any of the $x_i > y$, for ...
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198 views

If $20 $ random numbers are selected independently from the interval $(0,1) $ probability that the sum of these numbers is at least $8$? [closed]

If $20 $ random numbers are selected independently from the interval $(0,1) $ what is the probability that the sum of these numbers is at least $8$? I tried to take this question https://math....
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1answer
47 views

Is the sum of 3 bits a linearly separable task?

In other words can a linear classifier learn to correctly assign a class (label 0 to 3) for an input of 3 bits? Intuitively this cannot work, since the half-adder circuit contains an XOR block, which ...
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28 views

Sum of a number of shifted exponentially distributed random variables

I know that the sum of $k$ independent exponentially distributed random variables each with density function: $$\displaystyle \lambda\,{{\rm e}^{-\lambda\,x}}$$ has an Erlang distribution: $$\...
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1answer
108 views

Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables

Peggy Strait, 1974, Pacific Journal of Mathematics ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&...
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132 views

What is an intuitive explanation for Q90 (X+Y) > Q90(X) + Q90(Y) in fat-tailed variables. Non Subadditivity

In a business situation, management keeps a reserve of money for a 'rainy day' just in case costs are more than expected. The 90th percentile ($Q_{90}$ in the following) might be an indicator of how ...
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48 views

Concentration of sum of geometric random variables taken to a power

I am interested in techniques for showing the concentration of sum of $n$ iid geometric random variables $X_1, X_2, \cdots, X_n$ (number of trials until success), say with success probability $p = 1/2$...
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28 views

Variance of ratio of sums

How do I compute the variance of a ratio of sums ? $$ Var\big(\frac{\sum_i X_i}{\sum_j Y_j}\big) $$ I have 2 datasets $X=(X_1,...,X_n)$ with $Y=(Y_1,...,Y_n)$ that I need to compare, and estimate the ...
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1answer
45 views

Conditional expectation, conditional on sum of weighted average of two iid RVs

I have an arbitrary distribution $F$, and two variables $z, x \sim F$. I only observe the weighted average $y = \alpha z + (1 - \alpha) x$. Conditional on $y$, what is the expected value of $z$? I ...
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28 views

Mean number of throws to exceed a threshold [duplicate]

Say that you have a die with n faces, and you need to throw the die until the sum of your results exceeds a given threshold. What is the average number of throws needed? I think that to compute that ...
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1answer
16 views

Sum of estimated costs for uncertain events

I have a number of possible events $e$ with a probability $p_e$ of the event occuring and a cost estimate should the event occur (if it doesn't occur the cost is 0). The probability for each event is ...
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33 views

What is the expectation of $\left\langle (n \bar{y})^4 \right\rangle$, if $y_i \sim \mathcal{N}(\mu,\sigma^2)$? [duplicate]

Let $y_i \sim \mathcal{N}(\mu,\sigma^2), \; i = 1,\ldots,n$ and $\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$, such that $n \bar{y} = y_1 + \ldots + y_n$. Then, we want to know what the expectation of $(n \...
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1answer
71 views

Fast Evaluation of a Double Sum

Let $q$ be a probability distribution on $\mathcal{X}$, $w$ be a nonnegative function from $\mathcal{X}$ to $\mathbf{R}$ which is bounded away from $0$ and $\infty$, and $s$ be a bounded function ...
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1answer
38 views

How can we decompose $\text{Var}[\sum_{i=1}^n\sum_{j=1}^m f(A_i,B_j)]$?

Formulas for decomposing the variance of a summation of random variables can be found on Wikipedia but what is the variance of a double summation of a function of random variables? That is, are there ...
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20 views

Sum of Poisson Random Variables with Different Time Units

Suppose that W = the number of women who enter a store in an hour, M = the number of men who enter a store in three hours, Z = W + M. Suppose you believe that W ~ Poi(100) (so you have that the ...
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30 views

Sum of Bimodal Distributions? [closed]

If I'm trying to estimate a the sum of a bunch of random variables, where each random variable is a bimodal distribution, how would i go about thinking or modeling what that looks like?
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22 views

Z-Score for Sum of Proportions?

First, please bear with me. I am not very savvy with statistics, so I may be mixing up terms or using things improperly. Second, I am dealing with statistics related to baseball, so let me explain ...
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60 views

Sum of IID normal variables with index following Poisson distribution

$X_1, X_2,\ldots$ are a sequence of independent normal random variables with mean 1 and variance 1. Calculate the variance of $X_1+X_2+X_3+\ldots+X_{N+1}$ where $N$ follows Poisson distribution with ...
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264 views

What statistical test can compare the sum instead of the mean?

I'm confused about if t-test can only test means (sum / sample size), or if it can test sums as well (not normalizing for sample size). Below is a passage from a trusted book. Note that even though ...
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1answer
59 views

Proportion of the sums or average proportion of the parts?

Here’s a curiosity that come up during a work discussion. While this example uses financial data, it has a statistical question at its heart. Consider the following table with a budget and actual ...
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1answer
36 views

I need to find the Z-score of a population to use as a cutoff point in order to reduce the value sum of numbers to a new sum [closed]

This may be tough to describe, but I'll give it a shot. I am setting up an analysis that produces a large set of numbers. Let's call this Analysis 1 (A1). If I run A1 and return 60 numbers with a ...
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2answers
422 views

Is there a statistical distribution whose values are bounded $[-1,1]$ and sum to 1?

The Dirichlet distribution contains values that are bounded $[0,1]\in \mathbb{R}$ and sum to $1$. Is there a parametric distribution or similar method whose values do the same but reach as low as $-1$?...
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72 views

Any known approximations of summing quantiles from joint (bernoulli / lognormal) distributions

This is my first post to this site! For an insurance-like scenario, I have several independent risks which I want to sum together and find a 95% percentile. Currently I do this by Monte Carlo but I ...
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1answer
112 views

How to calculate this dependent probability (marbles without replacement)?

I present the question in two steps: First: Let there be 100 bags. A person puts 5 marbles into 5 separate, randomly selected, bags. You are now to collect the contents of the bags, one by one. If you ...
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1answer
507 views

Expectation of sum of absolute values for correlated normal random variables

Let $x_1, x_2, \dots, x_{N}$ i.i.d. random variables $\sim \mathcal{N}\left(0,\sigma^2_x\right)$. Further, let $z\sim \mathcal{N}\left(0,\sigma^2_z\right)$, $z$ is independent from all $x_i$. We build ...
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1answer
71 views

X~Unif(0, 1) ; X1 + X2 + ... X6 = 1 ; Y = sum(X1...X6) ; VAR(Y) =?

Let $X_i$ ~ Unif(0, 1) s.t. $X_1 + X_2 + ... + X_6 = 1$ Let $Y = X_1 + … X_6$ What is $Var(Y)$? (Also the case when it's $X_n$) Purpose for the curious: I'm trying to rank confidence for softmax ...
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3answers
126 views

What does "The mean of the sum of N independent variables with the same distribution is N times the mean of a single variable" mean?

I have been reading a book about statistics for physicists and there was this line given: "The mean of the sum of N independent variables with the same distribution is N times the mean of a single ...
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1answer
19 views

Limiting distribution of infinite sparse sum

Let $N$ be a positive integer. I consider $N$ random variables $X_1^{(N)}, X_2^{(N)}, \dots, X_N^{(N)}$, all independent and identically distributed, each taking values $\pm 1$ with probabilities $p/(...
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1answer
50 views

probability distribution of a sum of random variables [closed]

Suppose we have a random variable $X$ $P[X=-1]=1/3$, $P[X=0]=1/3$ and $P[X=1]=1/3$ now let $Y=X^2$ we have $n$ independent realizations of $Y$ $(Y_1, Y_2,......, Y_n)$ what is the probability ...
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1answer
221 views

Distribution sum of correlated normal variables squared

I'm trying to deduce which distribution my data follows and how to estimate the parameters. I have four random variables $X_i \sim N(\mu_i,\sigma_i^2)$ where the means and variances are all different. ...
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2answers
238 views

What are continuous distributions that are additive and have finite support

I'm wondering what are continuous distributions that are additive and have finite support. Joint normal distribution is continuous, and is additive in the sense that if $X,Y$ are joint normal, then $X+...
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700 views

How to interpret sum of two random variables that cross domains?

suppose we have two discrete random variables: $X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution) $Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform ...
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1answer
132 views

Perform signed rank test where there are duplicates in data?

How do you perform a rank test when there are duplicates in the data? that is, we have a dataset with numbers $1,1,1,1,3,3,3,3,4,4,4,4.$ and another dataset also with duplicates.... Is the sum of ...
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1answer
207 views

How do I find the conditional distribution of a normal r. v. z, given that I know the sum of z and another normal r. v. x is greater than some value?

Suppose I have two independent normal random variables, $X$ and $Z$ with $\mu_x$, $\sigma^2_x$ and $\mu_z$, $\sigma^2_z$. Suppose I also know that $x+z\geq y$. How do I find the conditional ...
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1answer
51 views

Why is $\sum{(x_i-\overline{x})^2}$ = $\sum{(x_i-\overline{x})x_i}$ true? [duplicate]

I have seen this equality many times in books but I never found an explanatory derivation.
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39 views

When (if ever) is the sum of two dependent geometric RVs negative binominal?

Imagine you have two random variables $X $ and $Y$, you know $$ X \sim \text{Geometric}(p) \\ X + Y \sim \text{Negative Binomial}(2, p) $$ I am interested in what if anything can be said about the ...
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44 views

Independence of random variables and sums of random variables

I am seeking to find the joint distribution of X and Y. I have the marginal distributions of X and X+Y and they are independent. We have that $f(X=x,Y=y)=f(X=x,X+Y=x+y)$ which is equal to $f(X=x)f(X+...
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127 views

Convolutions of joint random variables

I have two discrete dependent random variables $X,Y$, where both $X$ and $Y$ can take values either $0$ or $1$. Furthermore, I know their joint distribution $f_{X,Y}(X,Y)$. Now let's say I have an ...
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2answers
286 views

Why is the sum of all the elements in a Gaussian-distributed list with zero mean not zero?

If I generate a list of elements which has a Gaussian distribution with zero mean: List = np.random.normal(0, 1, 500) my intuition (why is obviously wrong) tells ...
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1answer
245 views

Aggregation with an overlap: Dirichlet distribution

Suppose that we have $$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$ where $p_4=1-p_1-p_2-p_3.$ When we add random variables for example, $p_1+p_2$ and $p_3+p_4$, the resulting distributions ...
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1answer
64 views

residuals in the simple regression model

The residuals in the simple regression model have to sum up to 0?
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1answer
1k views

Linear regression $y_i=\beta_0 + \beta_1x_i + \epsilon_i$ covariance between $\bar{y}$ and $\hat{\beta}_1$

I am currently reading through slides from Georgia Tech on linear regression and came across a section that has confused me. It states for $$ y_i=\beta_0+\beta_1x_i+\epsilon_i $$ where $\epsilon_i \...
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2answers
51 views

How can I prove these propositions of infinite sum of random variables?

$x_1, x_2, x_3, ..., x_i, ...$ ~ $uniform(0, 1)$ The actual random variable is the following. $P_i = (1-x_1)(1-x_2)...(1-x_{i-1})x_i$ And the goal is proving these... $\sum_{i=1}^{n}P_i \leq 1$ If ...
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2answers
312 views

Does the sum of discrete uniforms converge to a discrete Gaussian?

Is there some analogous of the Central limit theorem for discrete uniforms and discrete normal distributions? To be more specific, let's say we have identical and independent random random variables $...
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2answers
76 views

Inferring random variables from their sum

Suppose I have a large set of receipts that list the items I bought, but only list the total cost. One day I might have bought Milk, Butter, and Eggs. A different day I might have bought Bread, Milk,...
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2answers
353 views

Deconvolution of the sum of three gaussian distributions

Consider the sum of three normal random variables: $ R_{i,j}=A_{i}+B_{j}+C_{i,j}\, $ where $ A_{i}∼N(μ_{A},σ_{A}) $ , $ B_{j}∼N(μ_{B},σ_{B}) $ and $ C_{i,j}∼N(μ_{C},σ_{C}) $ . Assuming $A$, $B$ ...
2
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1answer
53 views

Force sum of random varables to equal to 1 [duplicate]

Suppose I have 3 random variables, $X1, X2,X3$. Define $Z$ as: $Z=X1+X2+X3$ I want to force $Z$ to equal 1 for every "realization" of $X1,X2,X3$ ($X_i \sim Beta(a_i,b_i))$. As an example, let $X_i$ ...