All Questions
Tagged with variance probability
62 questions with no upvoted or accepted answers
5
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343
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Variance of quotient of Poisson random variable and sum of the Poisson sample
Let
$$Y_1\sim \operatorname{Poisson}(\lambda_1)\\Y_2\sim \operatorname{Poisson}(\lambda_2),$$ where $Y_1$ and $Y_2$ are independent, and $\lambda_1, \lambda_2>0$.
What is the variance of $$\frac{...
- 105
4
votes
0
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218
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Comparison of Difference of Expectations of Conditional Variances
I want to show (if possible) that
$$\mathrm{E}[\mathrm{Var(Y|X_1, X_2)}] - \mathrm{E}[\mathrm{Var(Y|X_1)}] \geq \mathrm{E}[\mathrm{Var(Y|X_1, X_2, X_3)}] - \mathrm{E}[\mathrm{Var(Y|X_1, X_3)}] \tag 1$...
- 113
4
votes
0
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703
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Expectation and variance of sample mean with random sample size
I have a question regarding sampling where the sample size itself is a random variable.
Say I have two sub-populations $A$ and $B$ from which I can sample a real valued random variable with ...
- 41
4
votes
0
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101
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Index of dispersion with approximate distribution
I have an unknown discrete probability distribution $D$ ($D$ is a probability mass function), defined on an interval $[a,b]$ ($a>0$) and an estimation $\hat{D}$ such that, for all $t\in[a,b]$,
$$(...
- 213
3
votes
0
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90
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Proof of integrated volatility
Let $X_i$ be a sequence of iid standard normal random variables, $\sigma:[0,1]\to\mathbb{R}_+$ a continuous function. Define $r_{n,i}\equiv\frac{\sigma(i/n)X_i}{\sqrt{n}}$. Show that:
$$
\sum_{i=1}^...
- 919
3
votes
0
answers
2k
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Variance of an unbiased estimator is 0 when the sample size goes to infinity
So I would like a proof for the following but I can't seem to do it myself.
I have a random variable $X$ and I draw $n$ samples($\{X_1, \ldots, X_n\}$) from it and I have
$$
Z_n = \frac{\sum_{i = ...
- 221
2
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0
answers
32
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Proportion of explained variance for a probability model(binary logistic regression)
in the article written by Frank Harell ,Statistically Efficient Ways to Quantify Added Predictive Value of New Measurements,(https://www.fharrell.com/post/addvalue/)
Harell is writing:
For a ...
- 1,035
2
votes
1
answer
239
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Will change in standard deviation impact covariance?
If we increase the degree of standard deviation of one variable, does it affect covariance of two variables?
Example, two variables are there, A & B, the covariance of A & B is 100, and the ...
2
votes
1
answer
225
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Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?
It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
2
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0
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56
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Variance of 2 Protocols: Sampling Coloured Balls with Dots
Suppose, we have an urn where each ball has one of $M$ colours and some balls have a dot. We would like to estimate the proportion $p$ of balls that have a dot. We have two experimental protocols:
We ...
- 21
2
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0
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234
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Calculate Variance from Dirichlet-like Distribution Empirically
I'm interested in the proportion of time that a sensor is in a particular state. The sensor tells me the amount of time that it's in each state, which I will denote by $X = \{ X_1, X_2, X_3\}$. I ...
- 737
2
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0
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87
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When is variance of sample maximum greater than unconditional variance?
Let $X_1$,...,$X_n$ be $n$ i.i.d. RVs with continuous distribution $F$. Further let $X_{(1)}$,...,$X_{(n)}$ be the associated order statistics such that $X_{(1)}<X_{(2)}<...<X_{(n)}$.
Under ...
2
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0
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130
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Existence of estimator that reaches Cramer-Rao bound
There is a well known classical result called Cramer-Rao bound:
https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound
Particularly, it is a lower bound for a variance of any unbiased estimate. ...
- 121
2
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0
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3k
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variance of multiple variables
Mean or $E(X)$ is linear, so it's valid to write $$E(x_1 + x_2 + x_3) = E(x_1) + E(x_2) + E(x_3)$$ But $Var(x)$ is not linear, so we write $$Var(ax_1 + bx_2 ) = a^2Var(x_1) + b^2Var(x_2) + 2ab\;Cov(...
- 223
2
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0
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204
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Variance of Distributions from the Exponential Family
I want to understand how the variance of an exponential family behaves. To take a very concrete example. Let consider the unit ball $B$ in d dimensions.
Consider the following distribution over unit ...
- 301
1
vote
0
answers
20
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central moments of random variable from _estimates_ of draws from the distribution function
I am trying to estimate the first two central moments of random variable $r$. The information I have about $r$ is a set of estimates $\hat{r}_i$ for $i \in \mathcal{I}$, each with corresponding ...
1
vote
0
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86
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How to find $\mathbb{E} \left[\frac{\bar{\mu}}{\bar{\sigma}^2}\right]$?
I asked the same question on math stacks: MathStacks:, and some user suggest to ask it here for better insight. So this question has found interest in many research problems, but there have been no ...
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1
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0
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26
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Probability that both the mean and sample variance are both covered by their respective confidence intervals?
I am given the question: "What is the probability that both the mean is in its confidence interval for confidence level a and the variance is in its confidence interval for confidence level a?&...
- 11
1
vote
0
answers
32
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Finding variance from normal distribution
Suppose $Z_1$ and $Z2$ ~$N(0,1)$
Let $X_1=2Z_1$ and $X_2=X_1+\frac{\sqrt{3}}{2}Z_2$
Let $Y_1=\sqrt{3}Z_1+Y_2$ and $Y_2=Z_2$
I understand I have to show the mean and variance for $X_1$ and $X_2$ should ...
- 183
1
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0
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20
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how do i empirically estimate variance of conditional normal distribution?
I've tried searching for this, but maybe I'm not using the correct search strings. suppose I have joint distribution $P(X_1,X_2)$ over 2 continuous random variables $X_1,X_2$ that I can sample from. ...
- 445
1
vote
1
answer
61
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If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$
If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$
I have no idea what to address with the conditional expectation part.
Thank you for any comments, someone has ...
- 331
1
vote
0
answers
130
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Variance of a vector-valued random variable along a unit vector
Let $X$ be a vector-valued random variable with variance $\mathbb{V}[X] < \infty$. How is the variance of $X$ along a unit-vector $\hat{v}$ defined? Can we say that in general it is $\hat{v}^\top \...
- 2,286
1
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0
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31
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Understanding Covariance after Variance (visually)
2 Points i understood from variance derivation-
A) For calculating Variance we do not subtract (or mod add), but rather sum squared all points' differences from the mean.
B) Variance of 1,2,3 will be ...
- 11
1
vote
0
answers
52
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Correlation Based Models vs Covariance Based Models
I am trying to better understand why some models are "covariance based" vs. why some other models are "correlation based".
1) For example, a Multivariate Normal Distribution ...
1
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0
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145
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Dirichlet distribution parameters from known variances
Let's assume, I know the variances of Dirichlet distribution parameters. Let these variances be:
$Var[X_1], ..., Var[X_n]$.
Is there a analytical solution to derive the parameter value alpha_i given ...
1
vote
0
answers
83
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Law of Total Variance Issue
The Law of Total Variance says:
if the variance of X is finite then $V(X) = E(V(X|Z)) + V(E(X|Z))$
Suppose $X\sim N(0,1)$, $Y\sim \text{Cauchy}(0,1)$, $X$ and $Y$ are independent.
Define $Z \equiv X + ...
- 151
1
vote
1
answer
59
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Binomial Distributions Problem
A casino customer bets on red at roulette (probability of success is 9/19). If the result is red, the client is given 3 dollars; but if she loses, she pays 3 dollars. The client plays until she has ...
- 11
1
vote
0
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79
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Bounds on distance between two independently variables drawn from the same distribution
Suppose $X_1$ and $X_2$ are iid from an arbitrary distribution with variance $\sigma^2$. How can we derive an upper bound for:
$$P(|X_1-X_2|\ge\delta)$$
One simple idea is Chebyshev's Inequality, ...
- 51
1
vote
0
answers
60
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Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$
Let $p \in (0,1)$ and $X$ be a random variable such that $P(X=a) = p, P(X=-b) = 1-p$
Show that if $Y$ is another random variable such that $E[X] = E[Y]$ and $V(X) = V(Y)$ then $P(Y \ge a) \le p$ and ...
- 71
1
vote
0
answers
114
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Variance of bivariate normal distribution plus normal distribution
Problem:
$W = -27 + 0.3X + 0.45Y + E$
The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
- 107
1
vote
0
answers
152
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Replacing summation by integral in classical variance of sum formula, is it possible?
It is well known that the variance of the sum of $x_1,...,x_N$ random variables is the sum of their variances plus twice their covariances:
$\text{Var} \displaystyle\sum_{i=1}^{N}x_i =\displaystyle\...
- 121
1
vote
0
answers
47
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Expectation and variance of a stochastic time process conditioned on its past
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
- 97
1
vote
0
answers
88
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What are meaningful ways to interpret Monte Carlo-simulated non-normal data?
My question relates to Confidence Interval (CI) calculation of Monte Carlo-simulated non-normal data
As answers and comments to that question show the confidence interval for the given distribution ...
- 113
1
vote
0
answers
79
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CLT and 2 variables
Okay so there are 2 variables $D_i$ and $V_i$. Now $D= D_1 + D_2 + ... + D_N$ and $V = V_1 +.. +V_N$
Now I know the relationship is such that $E[D_i - a*V_i] = 0$ and
$Var[D_i - a*V_i] = E[D_i]$ ...
- 489
1
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0
answers
26
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Time-partitions of sample size
I am struggling with explain something I read in a Whitepaper. The essence is as follows.
Let's begin with a random variable $X$ defined as number of events in an hours. Further, we assume that $X \...
- 11
1
vote
0
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1k
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Mean and variance of call center data
I have a fairly involved homework question, I was wondering if I could get some help.
There are two types of phone calls arriving at a switch, long-duration and short-duration. Each day the number of ...
- 133
0
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0
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39
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How to check the Variances between 2 estimators are same or not
Let say I have 2 batches of electric bulb from some manufacturing processes
First batch was run from 10 am to 2 pm (just assume). In this batch total $N_1$ number of bulbs are produced and among them $...
- 405
0
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0
answers
29
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Expected value of a decreasing function of two random variables
My question is exactly equal to the question posted at Expected value of decreasing function of random variable versus expected value of random variable with just one extra assumption: the two random ...
- 1
0
votes
0
answers
28
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Constrained Cholesky Decomposition
Suppose that I have an $(n\times 1)$ vector of random variables, $\varepsilon$. However, I know that $k$ linear combinations of $\varepsilon$ are 0. Specifically, I know that for a $(k\times n)$ ...
- 1
0
votes
0
answers
42
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Conditional Variance of $Z_i|\sum_i\beta_iZ_i$
Let's assume I have $K$ i.i.d. standard normal random variables $Z_1,...,Z_K$. Hence, I know that $V[Z_i] = 1$ and $E[Z_i] = 0$ for all $i\in K$. I am faced with computing the following conditional ...
- 1
0
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1
answer
101
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Variance of Multimodal Generalized von Mises Distribution?
How do you calculate the variance of a Multimodal Generalized von Mises (MGvM) distribution? Given its complexity with multiple modes and asymmetry, I'm looking for:
Any formula or method to calculate ...
- 113
0
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0
answers
25
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Question regarding probability and maximum possible variance
I have the following question:
Suppose we have a set of 10 numbers (1, 2, ... , 10), each with a certain probability tagged to it.
Is it true that the highest possible variance is achieved when 1 and ...
- 31
0
votes
0
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59
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mean and variance
Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
- 1
0
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0
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38
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What is n when computing the standard error or variance for a statistic computed per 1000?
Let's say we want to calculate the standard error for a statistic that proportion of heads per 1000 coin flips.
So let's say we flip a coin 200 times. We see heads 50 times.
$\hat{\mu}$, our estimate ...
0
votes
0
answers
77
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Maximize Variance of Linear Combination of Matrix Columns
Let $A$ be a $k \times 1$ random vector, and $\mathbf{A}$ be a $n \times k$ matrix of observations.
Letting $t \in \mathbb{R}^{k}$ be a vector of weights s.t. $||t||_2 = 1$, suppose we are interested ...
- 498
0
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0
answers
63
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Variance of $U= a \log (Z+b)-Z$ where $Z$ is the exponential random variable
Consider a random variable
\begin{align}
U= a \log (Z+b)-Z
\end{align}
where $a,b>0$ and $Z$ is an exponential random variable.
Question: Can we find the variance of $U$?
Things that I tried
...
- 205
0
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0
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33
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Condition for the asymptotic non-zero point estimation of the variance
we know that a condition for a non-zero point estimate of the variance for a finite sample is that there exist at least two integers $i,j$ such that $X_i\neq X_j$. In other words $\frac{1}{n}\sum\...
0
votes
0
answers
55
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How much compensation is needed to take on risk?
We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase the prize to 80,000 dollars? ...
0
votes
0
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160
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Taking multiple samples from the same population
I am new to stats and I am having difficulty understanding the variances for multiple samples taken from the same population.
Suppose the population weight of a group of men has mean 80 kg and ...
0
votes
0
answers
30
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When calculating Horwitz-Thompson estimator, is it correct to multiply the pairwise terms of the calculation by two?
I'm currently trying to learn how to calculate the Horwitz - Thompson estimator for population variances. Using this formula
$$ \hat{V}ar(\hat{\tau}_\pi)=\sum\limits_{i=1}^v \left( \dfrac{1-\pi_i}{\pi^...
