All Questions
Tagged with variance probability
232 questions
28
votes
3
answers
4k
views
Confidence Interval for variance given one observation
This is a problem from the "7th Kolmogorov Student Olympiad in Probability Theory":
Given one observation $X$ from a $\operatorname{Normal}(\mu,\sigma^2)$ distribution with both parameters unknown, ...
- 4,329
23
votes
2
answers
4k
views
Law of total variance as Pythagorean theorem
Assume $X$ and $Y$ have finite second moment. In the Hilbert space of random variables with second finite moment (with inner product of $T_1,T_2$ defined by $E(T_1T_2)$, $\Vert T\Vert^2=E(T^2)$), we ...
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22
votes
2
answers
2k
views
How do I analytically calculate variance of a recursive random variable?
Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $X$ be the number of prizes you get. What is its variance?
...
- 331
18
votes
1
answer
31k
views
Variance in estimating p for a binomial distribution
How can I calculate the variance of $p$ as derived from a binomial distribution? Let's say I flip $n$ coins and get $k$ heads. I can estimate $p$ as $k/n$, but how can I calculate the variance in that ...
- 628
14
votes
4
answers
6k
views
Meaning of "Overdispersion" in Statistics
I am trying to understand what "overdispersion" means in statistics.
Based on the Wikipedia page, "overdispersion" is defined as follows : "In statistics, overdispersion is ...
14
votes
2
answers
6k
views
Variance of maximum of Gaussian random variables
Given random variables $X_1,X_2, \cdots, X_n$ sampled iid from $\sim \mathcal{N}(0, \sigma^2)$, define
$$Z = \max_{i \in \{1,2,\cdots, n \}} X_i$$
We have that $\mathbb{E}[Z] \le \sigma \sqrt{2 \log ...
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13
votes
2
answers
329
views
Why aren't "error in X" models more widely used?
When we calculate the standard error of a regression coefficient, we do not account for the randomness in the design matrix $X$. In OLS for instance, we calculate $\text{var}(\hat{\beta})$ as $\text{...
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11
votes
4
answers
4k
views
Variance of resistors in parallel
Suppose you have a set of resistors R, all of which are distributed with mean μ and variance σ.
Consider a section of a circuit with the following layout: (r) || (r+r) || (r+r+r). The equivalent ...
- 235
11
votes
1
answer
2k
views
Likelihood of my friend being able to guess skittle taste
I'm preparing for a data science interview, and here's a question I encountered during my preparation:
Your friend claims he can tell the five colors of skittles apart by
taste alone. The probability ...
user239903
10
votes
3
answers
5k
views
Counterexample for the sufficient condition required for consistency
We know that if an estimator is an unbiased estimator of theta and if its variance tends to 0 as n tends to infinity then it is a consistent estimator for theta. But this is a sufficient and not a ...
- 351
9
votes
2
answers
3k
views
Is the variance of the multivariate folded normal distribution known?
The mean and variance of the folded normal distribution are known. Consider now the distribution of $(|x_1|, \ldots, |x_n|)$, where $\mathbb{x} \sim N(\mu, \Sigma)$. The mean of the multivariate ...
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8
votes
4
answers
3k
views
Is it possible for a distribution to have known variance but unknown mean?
Many tutorials demonstrate problems where the objective is to estimate a confidence interval of the mean for a distribution with known variance but unknown mean.
I have trouble understanding how the ...
- 208
8
votes
2
answers
1k
views
Is variance the area under the curve of the distribution of a population?
I am trying to understand what variance is, I already know the "official" definition
"Variance is the average squared deviations from the mean"
But I am trying to give it a visual ...
- 290
8
votes
3
answers
11k
views
Variance of sum of random number of random variables (Cambridge University Worksheet)
In the vein of my last question, I'm now at a roadblock on question 3 of this sheet:
http://www.trin.cam.ac.uk/dpk10/IA/exsheet3.pdf
(note: it's not my intention to ask every question I get stuck on ...
- 347
8
votes
1
answer
856
views
Origin of strange formula for equilibrium standard deviation
In the paper
M. Avellaneda and J. H. Lee, Statistical arbitrage in the U.S. equities market, July 2008,
in the Appendix on page 46, how does he get equilibrium standard deviation as following:
$$...
- 2,799
8
votes
2
answers
13k
views
Expected value of maximum likelihood coin parameter estimate
Suppose I have a coin toss experiment in which I want to calculate the maximum likelihood estimate of the coin parameter $p$ when tossing the coin $n$ times. After calculating the derivative of the ...
- 83
8
votes
1
answer
12k
views
Variance and expectation of dot product
I am wondering what is the $E[\textbf{a}\cdot \textbf{b}]$ and $var[\textbf{a}\cdot \textbf{b}]$
where $\textbf{a}, \textbf{b}$ are independent random vectors. That is as a vector whose elements are ...
- 103
7
votes
1
answer
5k
views
Variance of sum of dependent random variables
Can you guys help me prove the following:
$$
\operatorname{Var}\left[\frac{1}{m}\sum_{i=1}^my_i\right]=\frac{1}{m}(1-\rho)\sigma^2+\rho\sigma^2
$$
where the sampled predictions ($y_is$) have ...
- 133
7
votes
1
answer
716
views
Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$
How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
- 303
7
votes
1
answer
358
views
What's the maximum expectation of a conditional variance, $E[\operatorname{Var}(X+Z_1 \mid X+Z_2)]$?
Let $X,Z_1,Z_2$ be 3 mutually independent RV's, with $Z_1, Z_2$ assuming $N(0,1)$ distribution. $X$ is constrained to have unit 2nd moment, i.e. $E[X^2] =1$, but may take arbitrary distribution. The ...
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6
votes
3
answers
2k
views
Binomial distribution intituition for N
I am unable to convince myself intuitively as to why the variance of a binomial distribution increases with increase in n (number of trials). In general, I expect that as n increases, the distribution ...
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6
votes
1
answer
232
views
What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?
What is the meaning of the quantity:
$$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$
Is there, for example, a geometric explanation? Is there a term for it in statistics?
- 161
6
votes
2
answers
1k
views
Variance of the modulus of a random variable
Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. What is the upper-bound on the variance of $Y=\left|X\right|$?
My gut feeling says that $\operatorname{Var}(Y) \leq \operatorname{...
- 591
6
votes
2
answers
3k
views
Higher-dimensional version of variance
If $X$ is a real-valued random variable,
$$\mathbb{E}[X^2] - (\mathbb{E}[X])^2$$
is the variance of $X$.
Suppose now that $X$ is a random variable that takes values on $\mathbb{R}^n$. Consider the ...
- 6,738
6
votes
1
answer
434
views
Law of Total Variance
I trying to experiment with law of total variance in order to empirically recreate theoretical results.
In particular I am interested in verifying that:
$$
Var(Y) = E[Var(Y|X)] + Var(E[Y|X])
$$
Let's ...
6
votes
1
answer
458
views
How to estimate variance of sample variance?
Given an arbitrary sample, sample variance would be calculated. But how the variance of sample variance should be estimated? I tried to do some simulations using influence functions estimation methods....
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5
votes
1
answer
376
views
Large Numerical difference in variance calculation : Unable to decipher
For the below pdf, I've calculated variance by two methods and observe a large difference (2.1477 vs 2.9100). Wondering why is this difference right at the first decimal? Is it just loss of precision ...
- 205
5
votes
3
answers
2k
views
Variance of a function of a random variable as function of the original variable
Suppose $X$ a random variable and $Y=f(X)$ a function of this variable
I know I can write $\mathbb V(Y)$ as $\mathbb E(f(X)^2)-\mathbb E(f(X))^2$
I would like to know if it is possible to write $\...
- 51
5
votes
4
answers
6k
views
Practical meaning of expected value (mean value), variance and standard deviation?
I have a question about concepts:
Expected value (mean value) - $μ$
Variance - $σ^2$
Standard deviation - $σ$
What is the practical meaning of these common concepts of the probability theory and ...
- 379
5
votes
4
answers
2k
views
Iterated expectations and variances examples
Suppose we generate a random variable $X$ in the following way. First we flip a fair coin. If the coin is heads, take $X$ to have a $Unif(0,1)$ distribution. If the coin is tails, take $X$ to have a $...
- 477
5
votes
2
answers
170
views
Bounding the distance of empirical average from its expected value
Suppose we have three sequences of random variables, $(Y_n)_n$, $(W_n)_n$, and $(X_n)_n$ such that:
If $Y_n=a$, then $X_n=b$. If $X_n=b$, then $W_n=c$. That is
$$
1_{[Y_n=a]}\leq 1_{[X_n=b]}\leq 1_{[...
- 935
5
votes
1
answer
2k
views
Computing variance of squared difference of i.i.d. uniform random variables
When $U$ and $U'$ are two i.i.d. uniformly distributed random variables on $[0, 1]$, show that $$\mbox{Var} \left( (U-U')^2 \right) = 0.04 $$
I tried plugging in the formula $\mbox{Var}(U^2)=E(U^4)-E(...
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5
votes
2
answers
1k
views
How is this minimum variance worked out for this importance sampling estimator?
I was stuck with the function 17.13 in the open source book deep learning on page 590.
For short, the question is that,
For the importance sampling estimator:
$$\hat s_q = \frac{1}{n}\sum_{i=1, x^{i}...
- 6,912
5
votes
1
answer
541
views
Expected value and variance of moving a token on a cartesian plane based dice rolls
A fair four-sided die has its sides labeled U, D, L, and R, respectively. A token is placed at (0, 0) on the Cartesian plane and the die is then rolled repeatedly. After each roll, the token is moved ...
- 55
5
votes
2
answers
3k
views
What is the variance of the difference of two random-variable indicators with a chance of intersection between events?
Let $X$ be an event whose probability P($X$) = $p$ and let $Y$ be an event whose probability is P($Y$) = $q$. The probabilit$Y$ of $X$ intersection with $Y$, $P(X \cap Y)$ = $r$.
$I_X$ is the ...
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5
votes
1
answer
3k
views
How do we derive that $S^2$ is chi-squared distributed (with $n-1$ df)?
The claim is that $$(n-1)S^2/\sigma^2$$ is chi squared distributed with degrees of freedom $n-1$.
$(n-1)S^2/\sigma^2$ can be written as $$\sum_i^n \left(\frac {x_i-\mu}{\sigma}\right)^2-\left(\frac {...
- 2,987
5
votes
1
answer
862
views
Interpretation of conditional variance of estimator of intercept in linear regression
$Y_i=a+bX_i+e_i$. $Y_i$ and $X_i$ are scalar r.v. We have,
$$
V(\hat b|X)=\frac{\sigma^2}{n\left(\bar{X^2}-\left[\bar{X}\right]^2\right)}
$$
and,
$$
V(\hat a|X)=\frac{\sigma^2 \bar{X^2}}{n\left(\bar{X^...
- 359
5
votes
1
answer
1k
views
Max and min variance of the integral of a stationary stochastic process
Let $X(t)$ be a stationary stochastic process with mean $\mu$, variance $\sigma^2$ and correlation function $\rho(t_1-t_2)$. Let the integral of a stochastic process be:
$$I = \int_0^L X(t) \, dt$$
...
- 1,235
5
votes
1
answer
107
views
How is this formula for variance derived?
There's a formula for the variance of the traffic flow between A and B, calculated from sample data, quoted in the UK's Traffic Appraisal Manual. No proof is given and part of me really wants to know ...
- 51
5
votes
1
answer
1k
views
Variance of a product of Bernoulli with another distribution
I have a distribution X, now I play the following game:
I toss a coin, if it falls on a head, I get nothing, if it falls on tails, I get a prize drawn from X distribution. I play the game N times.
My ...
- 569
5
votes
0
answers
343
views
Variance of quotient of Poisson random variable and sum of the Poisson sample
Let
$$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
5
votes
0
answers
102
views
Variance of coin tosses conditioned on their sum [closed]
Problem setting
Let $X_1,X_2,\cdots,X_n$ be tosses of coins (heads = 1, tails = 0) with mean $\mu_1, \mu_2, \cdots,\mu_n$. Let $S(\vec{X})=\sum_{i=1}^n X_i$ and denote variance of the random variable ...
- 591
4
votes
3
answers
448
views
How to properly visualize the change of variance of a bivariate Gaussian distribution cut sliced along a fixed variable?
I'm trying to understand the basics of Gaussian Distribution. I struggle to visualice how the variance of the conditional probability of say P (Y|X) changes when X is fixed (given X and Y have a joint ...
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4
votes
2
answers
30k
views
How to know when two random variables are independent?
A pair of r.v (X,Y) is equally likely to be of any of these pairs of values $(0,1), (1,0), (-1,0), (0,-1)$. Both X and Y have mean = 0.
$E[XY]=$?
Is $Var(X+Y) = Var(X) + Var (Y)$?
I know that if X ...
- 55
4
votes
4
answers
3k
views
Odds of a number appearing only one time in 500 spins on roulette
I recently observed a roulette wheel where one number appeared only once in five hundred spins. This is an American roulette wheel with 0 and 00, so the odds of any number hitting should be 1/38.
I ...
- 41
4
votes
2
answers
279
views
Variance of random variables involving two independent standard Normals
Let $X$ and $Y$ be two independent standard Normal variables. Let $M := \max(X, Y)$ and $L := \min(X, Y)$. It is given that the covariance between $M$ and $L$ is given by $\text{Cov}(M, L) = 1 / \pi$ ...
- 1,279
4
votes
1
answer
1k
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First two moments of a quadratic form in which the vector and matrix are random (though independent)
$\DeclareMathOperator\tr{\mathrm{tr}}$Let $x$ be a standard normal $p$-variate random variable, which is independent of a symmetric positive definite random matrix $Y$. I would like to compute the ...
- 115
4
votes
2
answers
72
views
How $Var[e^{\frac{-1}{X+a}}]$ varies with $n$ where $X \sim Bin(n,p)$?
I have a binomial random variable $X \sim Bin(n,p)$. I am interested in the variance of a function $f(X)$ given by :
$f(X)=e^{\frac{-1}{X+a}}$. Here $a>0$.
Specifically, I would like to know how $...
- 224
4
votes
2
answers
121
views
Extreme value distribution with unknown variance
Let $\{X_1,\ldots,X_n\}$ be a sequence of r.v. such that $X_i\sim N(0,\sigma^2)$.
It is usually stated in Extreme Value Theory textbooks that (for suitably chosen $a_n$ and $b_n$)
$$\mathbb{P}\left(\...
- 1,385
4
votes
2
answers
178
views
Finding the variance of a stochastic process
This is part 2 of this question Calculate the mean and variance of a stochastic process?
For the Polya Urn problem, I am trying to understand why the ratio of the variance is:
$$\operatorname{Var}(X_n)...
- 131