Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [variance]

For questions regarding the variance of a random variable in probability, as well as the variance of a list or data in statistics.

In probability and statistics, variance is a measure of spread among the possible values of a random variable or a list of values.

More information can be found here.

2532 questions
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Difference between Variance and 2nd moment

I understand that $Var(X) = E(X^2) - E(X)^2 $ And that the second moment, variance, is $E(X^2)$ How is variance simultaneously $E(X^2)$ and $E(X^2) - E(X)^2$?
slimydummy
  • 253
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5 answers

Variance of sine and cosine of a random variable

Suppose $X$ is a random variable drawn from a normal distribution with mean $E$ and variance $V$. How could I calculate variance of $\sin(X)$ and $\cos(X)$? (I thought the question was simple and tried to do a search, but did not find any good…
Hùng Phạm
  • 163
5
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1 answer

What does "$\Sigma^{\frac12}$" mean in the context of variational auto encoders?

I am studying about Variational Auto Encoders. I found the following equation. $$\mu +\Sigma^\frac12\odot\epsilon$$ I can imagine that $\Sigma$ is related to variance. What does the $\Sigma^{1/2}$ mean? And how is the notation called? I can not…
jef
  • 161
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2 answers

How to calculate estimator variance?

Is there a quick way to calculate this? $$\operatorname{Var}\left[\frac{1}{n-1} \sum_{i=1}^{n-1} (X_{i+1} - X_{i})^2\right]$$ $X_{i+1}$ and $X_{i}$ ~ $N(\mu, \sigma^2)$ I know that for $\operatorname{Var}[S^2]$ where $S^2$ is the sample variance we…
Mia
  • 339
3
votes
2 answers

Decomposition of variance

Suppose X is a continuous random variable that can take any value between plus and minus infinity. Furthermore, suppose A is a random variable capturing those events where X is below 0, and B is a random variable capturing those events where X is…
user120911
  • 439
2
votes
1 answer

Sample variance equation not producing right estimate of population variance?

Given a population of [7, 14, 21, 28], I am sampling two data points at a time, calculating the (unbiased) sample variance, averaging the results, and expecting the mean of the results to be an estimate of the population's variance (61.25). But I'm…
WalksB
  • 123
1
vote
1 answer

How to find Probability Mass Function when computing Variance of a random variable?

I'm trying to understand an example on finding Variance of a random variable. See example here on StatLect. Here's the relevant excerpt where I'm stuck. Let $X$ be a discrete random variable with support $$R_X = \{0,1\}$$ and Probability Mass…
u243676
  • 115
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vote
1 answer

Estimate the variance of a population given only the means of independent subpopulations

Given a set of independent samples $x_{i,j}$ where $i$ ranges from 1 to $m$ and $j$ ranges from 1 to $n$, it is easy to estimate the variance of the underlying distribution using a formula like \begin{equation} \text{Var}(x) = \sum_{i,j} x_{i,j}^2 -…
Mike Izbicki
  • 875
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How to calculate the variance of a two-particle distribution function?

I have calculated a two-particle distribution function called $f(r)$ where $r$ is the distance between of two particle (also there are three parameters,$\alpha$, $\beta$ and $\gamma$ in its formula which I should calculate $f(r)$ for different…
Wisdom
  • 395
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2 answers

Should the unbiased estimator of the variance of the sample proportion have (n-1) in the denominator?

I know that the variance of the sample mean is: $$\frac{\sigma^2}{n}$$ And that the unbiased estimator for that expression is: $$\frac{\sigma^2}{n-1}$$ The variance of the sample proportion is: $$\frac{p(1-p)}{n}$$ Does an unbiased estimator of…
Magnus
  • 703
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0 answers

Question about Variance Needs Help!

I would love it if anyone can help me with this variance. I have $R = 3X + 3Y$, $Q = 3X - 3Y$. Can I do: $\text{Var}(R + Q) = (\text{Var}(3X + 3Y + 3X - 3Y) = \text{Var}(6X) = 36* \text{Var}(X)$? I think this is correct, however, I don't know the…
Big Boy
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1 answer

Averaging a list of variances

I have a list of means and variances. I want to average them both. For mean I simply put: $$\mu = \frac{\mu_1+\mu_2+\cdots+\mu_n}{n}$$ However, for the variance, I can't figure out to average $n$ variances. For only two I found this answer…
Meryem Janati
  • 19
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1 answer

How did the variance get calculated?

The Elm Tree golf course in Cortland, NY is a par 70 layout with 3 par fives, 5 par threes, and 10 par fours. Find the mean and variance of par on this course. Mean was calculated as follows: Mean = 70/18 = 3.8888 Variance was found to be: second…
Kurapika
  • 175
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1 answer

Finding the unbiased estimator of variance

The information that I've been given: After a chemical spillage at sea, a scientist measures the amount, x units, of the chemical in the water at 15 randomly chosen sites. The results are summarized in the form ∑x=18 and ∑x^2=28.94. Before the…
Math Comorbidity
  • 137
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1 answer

How to compute the variance of $\Phi(Z-c)$ where Z is standard normal distributed

I want to compute the variance of $\Phi(Z-c)$ where Z is standard normal r.v. , $\Phi$ is the standard normal CDF, and c is a constant. I tried to start with the expectation but don't know what to do next $$E[\Phi(Z-c)] = \int \Phi(z-c) \phi(z) dz…
Bowen Deng
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