# Questions tagged [sum]

The sum of two or more random variables.

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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 $\sum_{i=1}^{K} X_i =... 1answer 29 views ### How to apply Lyapunov CLT to data I have a situation where I have around 30 classes of variables with different means and variances (though the means aren't too far from eachother; think 4-7) and that the distributions are right ... 1answer 183 views ### 2 approaches for Monte-Carlo : weighted sum of$\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$, can I write the 2 ... 0answers 50 views ### Probability that any element of a random unit-length vector is large [closed] Given a vector$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 ... 3answers 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.... 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 ... 0answers 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: $$\... 1answer 108 views ### Expectation of Maximum and Minimum of Partial Sums of Normal Random Variables Peggy Strait, 1974, Pacific Journal of Mathematics ON THE MAXIMUM AND MINIMUM OF PARTIAL SUMS OF RANDOM VARIABLES Gives a nice result (4.3) and (4.4) in terms of "standard normal random variables&... 2answers 132 views ### What is an intuitive explanation for Q90 (X+Y) > Q90(X) + Q90(Y) in fat-tailed variables. Non Subadditivity In a business situation, management keeps a reserve of money for a 'rainy day' just in case costs are more than expected. The 90th percentile (Q_{90} in the following) might be an indicator of how ... 0answers 48 views ### Concentration of sum of geometric random variables taken to a power I am interested in techniques for showing the concentration of sum of n iid geometric random variables X_1, X_2, \cdots, X_n (number of trials until success), say with success probability p = 1/2... 0answers 28 views ### Variance of ratio of sums How do I compute the variance of a ratio of sums ?$$ 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 ... 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 ... 0answers 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 ... 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 ... 0answers 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 \... 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 ... 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 ... 0answers 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 ... 0answers 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? 0answers 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 ... 0answers 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 ... 0answers 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 ... 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 ... 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 ... 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?... 0answers 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 ... 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 ... 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 ... 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 ... 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 ... 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/(... 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 ... 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. ... 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+... 4answers 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 ... 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 ... 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 ... 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. 0answers 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) $$I am interested in what if anything can be said about the ... 0answers 44 views ### Independence of random variables and sums of random variables I am seeking to find the joint distribution of X and Y. I have the marginal distributions of X and X+Y and they are independent. We have that f(X=x,Y=y)=f(X=x,X+Y=x+y) which is equal to f(X=x)f(X+... 0answers 127 views ### Convolutions of joint random variables I have two discrete dependent random variables X,Y, where both X and Y can take values either 0 or 1. Furthermore, I know their joint distribution f_{X,Y}(X,Y). Now let's say I have an ... 2answers 286 views ### Why is the sum of all the elements in a Gaussian-distributed list with zero mean not zero? If I generate a list of elements which has a Gaussian distribution with zero mean: List = np.random.normal(0, 1, 500) my intuition (why is obviously wrong) tells ... 1answer 245 views ### Aggregation with an overlap: Dirichlet distribution Suppose that we have$$(p_1,p_2,p_3,p_4)\sim Dirichlet(a_1,a_2,a_3,a_4),$$where p_4=1-p_1-p_2-p_3. When we add random variables for example, p_1+p_2 and p_3+p_4, the resulting distributions ... 1answer 64 views ### residuals in the simple regression model The residuals in the simple regression model have to sum up to 0? 1answer 1k views ### Linear regression y_i=\beta_0 + \beta_1x_i + \epsilon_i covariance between \bar{y} and \hat{\beta}_1 I am currently reading through slides from Georgia Tech on linear regression and came across a section that has confused me. It states for$$ y_i=\beta_0+\beta_1x_i+\epsilon_i$$where$\epsilon_i \...
$x_1, x_2, x_3, ..., x_i, ...$ ~ $uniform(0, 1)$ 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 ...