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.

1,172 questions
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We flip a coin until a tail or five heads in a row occur. What is the number of expected flips?

We flip a coin until a taild or five heads in a row occur. What is the number of expected flips? I have tried to solve this by first defining 2 random variables: X ...
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Proof of Variance of the Irreducible Error

In Introduction to Statistical Learning, given the general form of a quantitative response between a set of predictor variables and a target variable $$Y=f(X)+\epsilon$$ and the general form for a ...
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Calculating Expectation and Variance

I think I have an idea of how to solve these problems, but I keep getting stuck. Any help would be greatly appreciated! Also, sorry I know I messed up the formatting with the exponents in the problems....
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Delta Method: Estimate the Variance of $T$

Let $X = (X_1,\ldots,X_n)$ be a random sample, where $X_1 \sim \mathrm{Bern}(p)$. Let $\lambda = e^p$. Question: By law of large numbers, $T=e^{(\bar{X})}$ is a consistent estimator for $\lambda$, ...
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Minimum variance portfolio problem

So the question asks: There are N (N > 1) stocks with the same variance $σ^2$ and the same pairwise correlation coeﬃcient γ (i.e. $c_ij$ = γ for all i = j. γ is a given constant such that 0 ≤ γ < ...
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confused about meaning of a expectation of a function

https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff#Derivation well,in the "Derivation" part of the wiki link. i don't figure out why $E(f)=f$, does it imply that the function $f$ is ...
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Is this a special probability distribution?

Does the distribution function: $\frac{1}{\theta}e^\frac{-y}{\theta}$ Have a special name? If not, how can I find the variance? I keep running into a dead end when I try.
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Recalculate Standard Deviation

I know how to calculate a standard deviation when given a set of values. You must calculate the mean, then square the difference of each value between the mean to get the variance. Lastly, the ...
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I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal N(\... 1answer 29 views Difficulty in finding marginal distribution Let$X=(X_{1},X_{2})$have joint pdf.$$f(x_{1},x_{2})=\begin{cases}\frac{e^{-\frac{x_{2}^2}{2}}}{x_{2}\sqrt{2\pi}},\ &\text{if}\ 0<|x_{1}|\le x_{2}<\infty.\\0,\ &\text{otherwise} \end{... 1answer 499 views Sample variance: degree of freedom argument In sample variance we divide by n-1 and not n. I know a couple of arguments for this - one is that this is sort of a normalization to ensure that the expected value of sample variance is equal to ... 2answers 242 views Find \text{Var}(N) where P(N = n|Y = y) is \text{Possion}(y); Y is a gamma with parameters (r,\lambda) The question is as follows: Suppose that the conditional distribution of N, given that Y = y, is Poisson with mean y. Further suppose that Y is a gamma random variable with parameters (... 1answer 713 views The symbols for variance and covariance I learned to denote the variance of x as \sigma_x^2, and the covariance of x and y as \sigma_{x, y}. The covariance of x and x is then \sigma_{x, x}, but because that it just the ... 1answer 2k views What does “variance unity” mean?: “A normal distribution with mean \mu and variance unity”. I've just come across the sentence: "Suppose you have the sample \ldots from a normal distribution with mean \mu and variance unity." What does variance unity mean? Variance equals 1? I can't ... 0answers 289 views Calculating the variance of an average of N iid random variables I'm having some problems proving that the variance of an average of N iid random variables is equal to \frac{1}{N}\text{Var}[X_1], where X_1 is one of the considered random variables. Formally, ... 0answers 25 views Deriving variance In kinetic methods of analysis, the rate of appearance of products is often used to infer the initial concentration of a reactant or substrate. The problem with this method is that often the rate ... 0answers 92 views Variance of the sum of a random subset [closed] Let A be the group of whole numbers in the range [0, n). We choose uniformly at random from all subsets of size k (0 < k < n). The mean of the sum of the subset is \frac{n-1}{2} k. ... 2answers 6k views Variance of Negative Binomial Distribution (without Moment Generating Series) Given the discrete probability distribution for the negative binomial distribution in the form$$P(X = r) = \sum_{n\geq r} {n-1\choose r-1} (1-p)^{n-r}p^r$$It appears there are no derivations on ... 1answer 1k views Random Variables, Minimize Variance The variance of X_1, X_2 are 1 and 4, and the correlation coefficient p=-0.3 1)Calculate the variance of Z_1 = 2X_1+X_2 2)Calculate the value of a that minimizes the variance of Z_2 = aX_1+(1-... 0answers 425 views Expected value of mean squares. ANOVA with one factor currently i'm trying to find the expected value of the mean squares in an ANOVA. However, I think I have a mistake because the latter term is too long and confused me a little. Any help is welcome. ... 0answers 356 views The variance of a sum of random vectors There are n vectors each containing exactly q random variables as elements. Each vector is denoted I_k. Each variable within the vector has its own (normal) probability distribution, and the ... 1answer 2k views Calculating the standard deviation involving a moment generating function An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generation function:$$M_X(t)=\frac{1}{(1-2500t)^4}.$$Calculate the standard deviation ... 1answer 17 views Variance - particular outcomes of X If x_i is a particular outcome of the random variable X, and \mu is the mean of the distribution of X, then is it true that E[(x_i-\mu)^2] = Var(X)? Maybe I should ask a different question: ... 1answer 4k views Finding the expectation and variance from a probability generating function I need some help with the following question. I managed to get the p.g.f., and can get the expectation and variance in the normal ways, but need a helping hand in deducing them through the use of the ... 2answers 3k views How does Variance become an Autocorrelation Function? "For a Gaussian stochastic process X=\{X(t)|-\infty<t<\infty\} with mean function \mu(t)=0 for all t, its autocorrelation function is$$ E(X(t)\cdot X(s))=R(h)=\max(0,1-|h|), h=t-s. $$... 2answers 291 views N is a Poisson distribution with mean 4. Find \operatorname{Var}(N\mid N \geq 4) You are given that N has Poisson distribution with mean 4. Find \operatorname{Var}(N\mid N \geq 4) I tried to use the definition of variance, where \operatorname{Var}(X) = E(X^2) - E(X)^2 ... 0answers 162 views Monte Carlo integration and variance With the monte carlo integration of a function f(x), what do they mean with the variance? Is it the variance of the function we want to integrate? I = ∫^{\inf}_{inf} f(x)p(x) dx (with p(x) some ... 1answer 21 views Simple variance question: variance of the whole expression when one variable is random A very simple question about the variance! I'm interested in the variance of an expression with a random variable inside it. For an expression of this form: a = \sum_{i=1}^{m}(x_i \cdot y_i - z_i)... 2answers 279 views Variance of random walk model? I'm taking my second term of statistics, and I find myself obsessed with an unnecessary detail...again. As follows:$$Y_t=\rho Y_{t-1}+u_t$$That is to say, we are working with a time series. We ... 1answer 164 views How do you find variance? Let \{X_n\} be a sequence of independent identically distributed random variables with mean \mu and finite variance.$$T_n= \binom{n}{2}^{-1} \sum_{1\leq i< j\leq n} {X_{i}X_{j}} $$I have ... 1answer 3k views Variance of a normal distribution for coin toss. I have difficulties constructing the normal distribution for (20) coin tosses. (Don't ask why, but I never had probability in school.) What is the probability of getting at most 12 heads out of 20 ... 1answer 213 views Mean of the deviations from the mean I ve been struggling to understand the below problem.If I could get help with this problem, it would be greatly appreciated. Consider the data y, y+a, y+2a,….,y+na and the deviations of these ... 2answers 29 views Three random variables equation Given the random variables X,Y,Z have the same distribution and fulfil the following equalities:$$Var(X+Y+Z)=21,Cov(X,Y)=Cov(Y,Z)=Cov(Z,X)=1$$Find VarX and Var(X+Y). I am lost with this ... 3answers 538 views The relationship between sample variance and proportion variance? I'm trying to see the relationship between the sample variance equation \sum(X_i- \bar X)^2/(n-1) and the variance estimate, \bar X(1-\bar X), in case of binary samples. I wonder if the ... 1answer 79 views Calculate the variance and expectection of \hat{y} in a linear regression model I have the following linear regression model$$y = \beta_0 + \beta_1 \cdot 40$$where$\beta_0 = 11.1317$,$\beta_1 = 1.01$, and$40$is simply the value of the predictor variable (I guess). As you ... 2answers 77 views Finding Expectation and Variance of$X_1$and$X_2$Let$X_1$and$X_2$be random variables such that$E(X_i)=μ_i$and$Var(X_i)=α_i^2$. A. Find$E(X_1+X_2)$and$E(X_1-X_2)$in terms of the μ's and α's. B. Suppose that$E(X_1X_2)=α$. Find$Var(X_1+X_2)...
Here is a picture of my problem Basically, given that $R_X(t) = \log(M_X(t))$, I need to show that $\text{Var}(X) = R′′(0)$. As an attempt, I know that $\text{Var}[X] = E[X^2] - (E[X])^2$ and that \$...