I think this chain of reasoning contains a bit of confusion between mean and median. A p-value of 0.5 should be expected at the theoretical median (i.e. $F^{-1}(0.5)$), not the expected value. From ...

Given that car model is a categorical variable, replacing it with some numerical variables such as horsepower or mpg can have advantages depending on the scope of your prediction model. If you only ...

$N$ containts the realisations of 10000 random variables, each with a different mean. Therefore, the observed deviations from the average are higher. Same goes for $M,K$. If you calculate sd(N-U), ...

Consider a linear model $Y = X\beta + \varepsilon$ with $p$ variables and $n$ observations, $p>n$. Assuming the variables are not linear dependent, i.e. the matrix $X$ has rank $n$, $Y$ can be ...

Likelihoods express the probability, that the observed event (two heads up) occurs given some parameters ($p = 1$). If you have a coin that always lands on head, two heads in a row have a probability ...

First, $G(0) = 0$. For $y > 0$, $P(x^{-2} \leq y) = P(x^2 \geq y^{-1}) = P(x \geq y^{-0.5}) + P(x \leq - y^{-0.5})$. You just missed one change from $\leq$ to $\geq$ in the middle.

The answer is no, here is a counter example: $P(y = 1) = P(y = -1) = 0.5, ~ x = z y$ with $P(z = 1) = P(z = 2) = 0.5$ and $z$ independent of $y$. Since $z = x/y$, $z$ and $x/y$ are perfectly ...

In your notation $B_2$ describes the difference between the effects of being female and being male. Everything else is in the intercept $B_0$. Consider this example: Assume a imaginary linear ...

Maybe your understanding is helped by rewriting a part of the formula as: $(x-\mu)^\top S^{-1} (x-\mu) = (S^{-1/2} (x -\mu))^\top S^{-1/2} (x-\mu)$ Then you can see that both $x$ and $x^\top$ are ...

summary(mod) gives you a nice overview over what's going on under the hood. Basicallly what happens is that the categorical variable is automatically turned into a number of binary variables, i.e. is ...

Are you allowed to use a package that provides you with independent normal distributed RVs? If not, the Box-Muller method (https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform) could be used for ...

This is a model decision you'll have to make. When fitting a linear model, you have to make assumptions about the true relationship between the predicted variable and the predicting variables. The ...

To answer the first part of your question: No variable is selected, if it is optimal to not change any element of $\beta$ away from zero, i.e. $y^\top y \leq (y-x_i \hat{\beta_i})^\top (y-x_i \hat{\... View answer 1 votes If I understand your question correctly, you have 1300 observations. For some of your outputs, 1295 of 1300 observation have the value zero and 5 obervations have a value$\neq$0. Treated ... View answer 1 votes These 4 plots show the probability densitiy functions of some distributions, therefore they do not need to be in the range of$[0,1]$. They are not cumulative distribution functions. View answer 0 votes If you don't have too many different values for the categorical variable, you could encode each value as a binary variable (e.g. for colour of car: one variable: 1 if car is green, 0 else). For these ... View answer 0 votes user20160 is right. Just to add a little technical help: you simulate data from a Gaussian distribution with covariance matrix$K$by calculating the Cholesky-Decomposition$K = L L^\top\$ and ...