All Questions
Tagged with variance probability
232 questions
2
votes
2
answers
911
views
MLE of variance is biased in a Gaussian distribution
Referring to: How to understand that MLE of variance is biased in a Gaussian distribution
at some point during calculation the formula of the sum of the expected value becomes a single expected value:...
- 33
1
vote
2
answers
360
views
Changing only one point of a discrete distribution to maximize variance augmentation
X has a discrete distribution with support $x1, x2, ...$ in $ {]}0,1{[}$. You have the right to change only one of the $xi$ to lead to the highest increase in variance (or, at least, a systematic ...
- 21
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
3
votes
2
answers
95
views
$X$ has distribution function $F(x) = e^{-e^{-x}}$. Justify that such a probability measure on $\mathbb{R}$ exists
How can I prove a probability measure exists? If $F(x) \rightarrow 1$ as $n \rightarrow +\infty$, does that mean $F(x)$ does exist? And how should I calculate $\mathbb{E}(F(X))$ and $Var(F(X))$?
- 31
2
votes
1
answer
499
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Buffon's Needle problem
So I'm working through some computational stats stuff from a free pdf of a book. Specifically I'm looking at their take on the classic Buffon's needle problem. The question has a theoretical part and ...
- 121
0
votes
1
answer
22
views
How the variance of a potential loss X has been derived
I'm studying Insurance and I have a question about how the variance has been computed in this example.
Imagine a case where an "agent" may suffer a loss, because of an event (an accident) occurring ...
- 155
0
votes
1
answer
39
views
Obtaining Negative Variance. What is the error?
Suppose a dice is thrown $8$ times and success is considered as obtaining either a $5$ or $6$. What is the variance of the number of successes?
Attempt: Let the indicator variable $X_i$ be $1$ when ...
- 223
3
votes
2
answers
3k
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Finding the maximum and minimum variance of the sum of two Bernoulli events?
You are guessing the contents of two envelopes. Let $U_i$ be the event that you guess correctly. Your probability of guessing correctly for each envelope is $P(U_1) = P(U_2) = 3/4$. $U_1$ and $U_2$ ...
0
votes
1
answer
588
views
Conditional variance of a random variable conditioned on its own value
Suppose that $X$ is a random variable. Does it hold that $\mathbb{V}ar[X|X]=0$? What is the proof/intuition behind this?
- 145
0
votes
1
answer
70
views
Compute Conditional Variance
Let the joint density $ f_{X,Y}(x,y)=\begin{cases} c(x^3+2xy),\ 0\le x,y\le 2\\
0, \text{ else}\end{cases}$
be given. I want to compute $Var(Y|X=1)=\int^\infty_{-\infty} (y-E(Y|X=1))^2f_{Y|X=1}(y)\,\...
- 279
1
vote
0
answers
8k
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How to calculate variance or standard deviation for product of two normal distributions? [duplicate]
For example if I have two multiplied distributions a * b:
...
- 111
0
votes
1
answer
181
views
Variance of scalar function of 2 random variables
Suppose I have a scalar function $g(X,Y)$, where $X$ and $Y$ are jointly distributed with pdf $p(x,y)$. I think the expected value of $g$ is given by
$$ \mathbb{E}[g] = \int_{-\infty}^\infty \int_{-\...
- 1
1
vote
1
answer
418
views
Get the new variance of the data [duplicate]
I got an initial mean $\mu_1$ and std $\sigma_1$ by sampling samples, these samples are generated by an unknown distribution and later I drop these samples. Then I sampled some samples and got the ...
- 1,391
13
votes
2
answers
329
views
Why aren't "error in X" models more widely used?
When we calculate the standard error of a regression coefficient, we do not account for the randomness in the design matrix $X$. In OLS for instance, we calculate $\text{var}(\hat{\beta})$ as $\text{...
- 64.8k
0
votes
0
answers
94
views
Conditional Covariance Problem
Suppose we have independent (not necessarily identical) normally distributed random variables X, Y. If we're given that, upon sampling each variable, X is some multiple a of Y (i.e. x = ay), what is ...
1
vote
3
answers
1k
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Calculate variance the right way with two random variables
I'm currently assigning a introductory stats class, and I just can't seem to find out when to use the different variance identities. I have provided an example of an assignment where I got it wrong, ...
- 13
2
votes
1
answer
225
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Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?
It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
7
votes
1
answer
716
views
Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$
How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
- 303
1
vote
1
answer
225
views
Variance and covariance inequality
Given a real-valued random variable $X$, is
$$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$
true?
Any pointers for how to tackle this problem would be immensely helpful.
- 549
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
0
votes
1
answer
37
views
Expressing as a probability density function [closed]
The measuring error x is a normal random variable. Variance of the error = 4. If distribution of x can be shown by a probability density function f(x), how would you find the analytical expression of ...
- 1
0
votes
1
answer
275
views
How to estimate the mean and variance of a Gaussian distribution variable? [closed]
I have two variables 2X and 0.5Y, both are independent and follows Gaussian distribution. How to estimate their mean and variance analytically? I want to know their individual mean and variance, then ...
0
votes
1
answer
117
views
Summation of two Gaussian distributed data with different coefficient of mean and variance
I need some help on Gaussian distribution. i have two dataset, both are identical and independent distributed, but having mean as 2μ_1 and μ_2, same scenario for the variance. How can I add them?
...
0
votes
1
answer
161
views
estimating the mean of constant + noise
(This is almost certainly covered in Statistics 101, but I missed that class..)
I have a real-world sampled signal $S[t]$ that is a constant $\hat{S}$ plus some noise $\epsilon[t]$. My goal is to ...
- 155
1
vote
1
answer
4k
views
Does the peak of a Normal Distribution mean anything? [closed]
What does the peak of a Normal distribution show? Let's say if I have a flat peak, does this mean I have a larger variance? What if I have a sharp peak?
For example,
Does the "blue distribution" ...
- 581
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 ...
- 113
1
vote
2
answers
545
views
Two distributions, same mean, different variance: Stochastic dominance for deviation from mean?
Say you have two (bounded) random variables, $X$ and $Y$, on the same discrete probability space, such that $E(X)=E(Y)$ but $Var(X) < Var(Y)$. Do I know that, for any $k \geq 0$,
$$
\text{Prob}(|X-...
- 33
1
vote
0
answers
114
views
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
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
0
votes
1
answer
133
views
Variance of linear combination of Normal distributions
A company that develops software received an order for a service to be performed within a week and, in order to decide on the profile of the team of programmers to be used, it should take into account ...
- 107
1
vote
0
answers
152
views
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
2
votes
2
answers
196
views
Variance being negative
Let $X$ and $Y$ have joint pdf such that
$$f(x,y) = 3e^{-3x-y}, 0 < x< \infty, 0< y< \infty.$$
(a) Show that $X$ and $Y$ are independent.
(b) Calculuate $Var(X)$.
In ...
- 21
3
votes
1
answer
4k
views
When is the variance of the sum of random variables greater than the sum of the variances?
My professor asked my class to 'qualitatively' analyze the two scenarios with the assumption that there is no previous knowledge held in the concept of covariance (as we have not covered that chapter ...
- 101
0
votes
1
answer
93
views
Probability - expected value
The random variable $X$ takes on values -2, 0 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find $\text{E}(X)$ and $\text{Var}(X)$.
Till this part, it was easy enough.
Then the question ...
user218970
2
votes
2
answers
51
views
Getting variance of function of two uniform RVs [duplicate]
Have two independent RV's $X$ and $Y$ sampled uniformly from $[0,1]$ and $C = (X-Y)^2$. Want $V(C$).
Rewrote as $V((X-Y)^2) = V(X^2) - 4V(X)V(Y) + V(Y^2)$ but that's too messy. Is it correct to write ...
- 153
2
votes
0
answers
56
views
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
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,
$$
V(\hat a|X)=\frac{\sigma^2 \bar{X^2}}{n\left(\bar{X^...
- 359
3
votes
1
answer
93
views
Quantifying explanatory potential
Suppose I have a random variable $T_j \sim Bernoulli(p_j)$ where $logit(p_j) = \theta x_j + \epsilon_j$ and where $\epsilon_j \sim \mathcal{N}(0,1)$. Suppose further that $\theta = 0.018$ and that I ...
- 1,200
2
votes
1
answer
72
views
How to set $\alpha,\beta$ such that $logit^{-1}(\alpha X_1+\beta X_2)$ has a mean of 0.4 with $X_1 \sim Bern(p)$ and $X_2\sim N(\mu,\sigma^2)$?
I am working in R, and am trying to generate values of
$$
logit^{-1}(\alpha X_1+\beta X_2)
$$
with $\alpha,\beta$ such that $logit^{-1}(\alpha X_1+\beta X_2)$ ...
- 4,260
-1
votes
1
answer
31
views
Variance from sampling from a collection of marbles
Suppose I have $N$ marbles, $k$ of which are black. Let $X$ be the number of black marbles obtained from randomly choosing $M$ ($\leq N$) marbles. What is the variance of $X$?
Obviously if $M=N$ then ...
- 101
0
votes
1
answer
213
views
Do the location and scale parameters always control the mean/median/mode and variance, respectively?
Does a location parameter always control the mean/median/mode values of a PDF?
Does a scale parameter always control the variance of a PDF?
If the answer to any of the above questions is yes, then ...
- 115
4
votes
2
answers
279
views
Variance of random variables involving two independent standard Normals
Let $X$ and $Y$ be two independent standard Normal variables. Let $M := \max(X, Y)$ and $L := \min(X, Y)$. It is given that the covariance between $M$ and $L$ is given by $\text{Cov}(M, L) = 1 / \pi$ ...
- 1,279
1
vote
0
answers
47
views
Expectation and variance of a stochastic time process conditioned on its past
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
- 97
1
vote
1
answer
7k
views
How do I calculate the standard error of the $\chi^2$ statistic?
Question: Suppose that you are testing the idd-ness of a random number generator, and you've done so with the permutation test and the monkey test. Both tests produce a $\chi^2$ statistic and a ...
- 235
2
votes
1
answer
56
views
A Doubt involving Variance Equation and Expectations
Consider the following,
$$
\begin{alignedat}{1}
\operatorname{Var}(X)&=E((X-E(X))^2)\\&=E(X^2)-(E(X))^2.
\end{alignedat}
$$
Since the expectation of a random variable is no longer random, let ...
- 143
3
votes
1
answer
281
views
How to compute variance of squared binomial RV?
If $T$ is distributed from a Binomial $\mathcal{B}(n,p)$ distribution, is there a simple way to compute the variance of
$$
\frac{T(n-T)}{n(n-1)}=\frac{\sum(X_i-\overline{X})^2}{n-1}
$$
where the $X_i$...
- 250
2
votes
0
answers
234
views
Calculate Variance from Dirichlet-like Distribution Empirically
I'm interested in the proportion of time that a sensor is in a particular state. The sensor tells me the amount of time that it's in each state, which I will denote by $X = \{ X_1, X_2, X_3\}$. I ...
- 737
5
votes
2
answers
1k
views
How is this minimum variance worked out for this importance sampling estimator?
I was stuck with the function 17.13 in the open source book deep learning on page 590.
For short, the question is that,
For the importance sampling estimator:
$$\hat s_q = \frac{1}{n}\sum_{i=1, x^{i}...
- 6,912
2
votes
1
answer
77
views
Given , $X$ is a standard normal R.V , I know $E[X|X>c]$ = $\frac{\phi(c)}{1 - \Phi(c)}$ , how do i derive a similar formula for $var[X|X>c]$
I can derive $E[X|X>c]$ = $\frac{\phi(c)}{1 - \Phi(c)}$ , using the trick $- \int \frac{d \phi(x)}{dx} = \int x \phi(x) dx$. How do I do a similar thing to derive $var[X|X>c]$.
- 31
1
vote
1
answer
91
views
From where term $\left(\frac{1}{n}+\frac{1}{m}\right)$ came in estimated variance of $\bar x - \bar y$
I encountered such a formula for pooled variance:
$$\frac{(n-1)s_x^2+(m-1)s_y^2}{n+m-2}\left(\frac{1}{n} + \frac{1}{m}\right)$$
Here we have two samples of the following sizes $n$ and $m$. $s_x, s_y$...
- 138