I have two variables from a multivariate standard normal distribution, which are highly correlated (say, r = .8). If I observe variable 1 to be 1.5, how can I predict the value (or the possible range of values) I would expect for variable 2?   I could do so using random draws of the distribution, but I think there should be an analytic way as well, probably making use of projections.
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I am trying to use an organism's length (x) to predict weight (y), but length and weight are highly correlated (r=0.95). Are there any pitfalls I should be aware of because estimates of the two variables are in a sense measuring the "same" thing, i.e., size? I would be happy for someone to point out that this question has been asked before, but I don't see it on this site.
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In regression, standardization is recommended in ordered to assess the relative importance of predictors. However there seems to be an assumption of normality? How would the interpretation work for predictors following different non normal distributions? It seems confusing...
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I want to run a regression with 4 to 5 explanatory variables, but I have only 15 observations. Not being able to assume these variables are normally distributed, is there a non-parametric or any other valid regression method?
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I have some pairs of datasets (n=200 or thereabouts), of samples which are non-negative and not normally distributed. I think these pairs of variables are related, probably linearly.   Calculating Spearman's rank correlation on these datasets gives some strange results. The correlation coefficients show that the pairs of variables are weakly, positively correlated (e.g. rho of around 0.4), but the p-values are very low (e.g. 4.1e-10).  My vague understanding of this is that the variables are weakly, positively correlated but the probability of unrelated variables producing the same correlation is very low. Does this mean we can be reasonably certain that a positive correlation exists or have I misunderstood?
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I've been given a problem where I have  $$ X \sim \mathcal{N}(\mu = 2, \sigma^2 = 9) $$  $$ Y \sim \mathcal{N}(\mu = 3, \sigma^2 = 4) $$  Their correlation is $ \rho_{XY} = 0.6 $. First I am asked for the mean and standard distribution of X+Y, which I found as  $$ \mu_{X+Y} = \mu_X + \mu_Y = 2 + 3 = 5 $$  $$ \sigma_{X+Y} = \sqrt{\sigma_{X}^2 + \sigma_{Y}^2 + 2 \rho_{XY} \sigma_x \sigma_y} $$  However, I am also asked what the distribution of $ X + Y $ is, what if $X$ and $Y$ are jointly normally distributed, and what if $X$ and $Y$ are not jointly normally distributed. I am unsure how I would go about determining what the distribution of $X + Y$ is, though I assume that    If $X$ and $Y$ are jointly normally distributed then $X + Y$ is normally distributed If $X$ and $Y$ are not jointly normally distributed then $X + Y$ is not normally distributed   How do I determine what the distribution of $X + Y$ is when I am not told whether they are jointly distributed?    Related  I read the following post, but wasn't sure how to apply what was said to my current problem, if I could at all.
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As I understand, we can get correlation by normalizing covariance using the equation  $$\rho_{i,j}=\frac{cov(X_i, X_j)}{\sigma_i \sigma_j}$$  where $\sigma_i=\sqrt{E[(X_i-\mu_i)^2]}$ is the standard deviation of $X_i$.  My concern is what if the standard deviation equals zero? Is there any condition that guarantees it cannot be zero?  Thanks.
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$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables.  I showed that $X$ and $Z= \dfrac{Y-\rho X}{\sqrt{1-\rho^2}}$ are independent standard normal variables.  Using this I need to show that   $$P(X &gt;0,Y&gt;0) = \frac14 + \frac{1}{2\pi} \cdot\arcsin(\rho).$$
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We have a bivariate normal process where $X, Y \sim N(0, \sigma)$, with no covariance.  (For convenience we can assert that $\sigma = 1$, or that we have a good estimate for its value.)  What is the distribution of the random variable $R(n) =  \sqrt{\overline{x_i}^2 + \overline{y_i}^2}$ &mdash; i.e., the Euclidean distance between the sample center of a n points and the true center (at the origin)?  Note that as defined:   $R(n) \ge 0$ $E[R(n)] \to 0$ monotonically as $n \to \infty$ The Rayleigh distribution gives us $R(1) = \sigma \sqrt{\pi/2} \approx \sigma 1.25$   Furthermore, based on a Monte Carlo simulation for $n \in [2, 25]$ with $\sigma = 1$:   Variance decreases monotonically as n increases Skewness appears constant across n at 0.63 Kurtosis appears constant across n at about 0.24   (This question is a simplified version of  that seems to have gotten derailed in complications.)
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If I have two variables $X$ and $Y$ which randomly take on values uniformly from the range $[a,b]$ (all values equally probable), what is the expected value for $\max(X,Y)$?
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X and Y are not correlated (-.01); however, when I place X in a multiple regression predicting Y, alongside three (A, B, C) other (related) variables, X and two other variables (A, B) are significant predictors of Y. Note that the two other (A, B) variables are significantly correlated with Y outside of the regression.   How should I interpret these findings? X predicts unique variance in Y, but since these are not correlated (Pearson), it is somehow difficult to interpret.  I know of opposite cases (i.e., two variables are correlated but regression is not significant) and those are relatively simpler to understand from a theoretical and statistical perspective. Note that some of the predictors are quite correlated (e.g., .70) but not to the extent that I would expect substantial multicollinearity. Maybe I am mistaken, though.    NOTE: I asked this question previously and it was closed. The rational was that this question is redundant with the question "". Perhaps I do not understand the other question, but I believe these are entirely separate questions, both mathematically and theoretically. My question is entirely independent from if "a regression is significant". Furthermore, several predictors are significant, while the other question entails variables not being significant, so I don't see the overlap. If these questions are redundant for reasons I do not understand, please insert a comment prior to closing this question. Also, I was hoping to message the moderator who closed the other question to avoid identical questions, but I couldn't find an option to do so.
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The following is the residuals vs predicted scatter plot for a regression model with two IVs.   Initially, I thought it was evidence of heteroskedasticity. But, I reasoned that although there is a visible pattern in the plot, the variance across different levels of predicted values is same.  To clear my doubts, I saved the standardized residual and predicted values and ran a bi-variate correlation test. The correlation is zero, as expected. I am, nevertheless, intrigued by this observation. Any idea why I might have obtained this pattern?  PS:- My dependent variable is a sum of two likert-type items (Response scale: 1-5). So, it's theoretical range is 2-10 and it has no absolute zero value.
123722
I am attempting to use simple linear regression to construct a 95% prediction interval for a continuous response variable (Y) using a continuous input variable (X). When examining my data, I realized that if I log-transform both X and Y, the assumptions of equal variance and normality are essentially met, but if I do not log transform both variables, they are not. I'm able to run all of the R code needed to get the prediction interval for the transformed data, and I've read online and elsewhere how to interpret this interval. However, is there a way to get a 95% prediction interval on the untransformed scale using the 95% prediction interval on the log-transformed scale? In other words, I'm interested in talking about the conditional mean response instead of the median response (which, according to what I've read, is how I must interpret my prediction interval). I could exponentiate the interval endpoints, but I'm fairly sure that's incorrect because E(e^{Y}) is usually not equal to e^{E(Y)}.
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Let  $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X&gt;Y)$. the answer is $0.2389$ but I do not know how to do it.
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Suppose I have a set of measurements taken at regular intervals, and I want to predict future values of one of those measurements. There are relationships between the variables being measured. For example, some variables may tend to rise together, or one variable might tend to fall when another rises. What are the state-of-the-art methods for this type of forecasting?
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Let $A,B,C,D$ be independent normal random variables distributed according to $\mathcal{N}(0,\sigma^2)$ and $E = A+B+C+D$.  The goal is to provide an optimal estimate of $\sigma^2$ given samples of $A,B,E$ but not $C,D$.  In this case optimal refers to minimising the root-mean square deviation of the estimate $\hat{\sigma}^2$:  $$RMSD(\hat{\sigma}^2) = \sqrt{\mathbb{E}\left[\left(\hat{\sigma}^2-\sigma^2\right)^2\right]}$$  I can think of a simple approach to estimate $\sigma^2$ given samples of $A,B$ which is to simply compute the sample variance. Also I know that the variance of $E$ is $4\sigma^2$ so I could compute the sample variance of $E$ and divide it by 4 to get an estimate of $\sigma^2$. However it should be possible to combine the information from both $E$ and $A,B$ to improve estimation $\sigma^2$. Could you outline the approach to use both sets of information?
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I am doing some calculations on different matrices (mainly in logistic regression) and I commonly get the error "Matrix is singular", where I have to go back and remove the correlated variables. My question here is what would you consider a "highly" correlated matrix? Is there a threshold value of correlation to represent this word? Like if a variable was 0.97 correlated to another one, is this a "high" enough to make a matrix singular?  Apologies if the question is very basic, I wasn't able to find any references talking about this issue (a hint towards any reference would be a big plus!).
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I have a basic question.  Say I have two random variables, $X$ and $Y$.  I can standardize them by subtracting the mean and dividing by the standard deviation, i.e., $X_{standardized} = \frac{(X - E(X))}{(SD(X))}$.  Is the correlation of $X$ and $Y$, $Cor(X, Y)$, the same as the covariance of the standardized versions of $X$ and $Y$?  That is, is $Cor(X, Y) = Cov(X_{standardized}, Y_{standardized})$?