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As Seyhmus mentioned, this should be $E[X^2] = \sum_r r^2 P(X=r)$. This is valid only for discrete random variables: the version for a continuous random variable with density $f(x)$ is $$E[X^2] = \i …

I assume you mean that you take the spot at the end of the road if it is available, and checking whether this is available takes no time; if this spot is not available, you go to the parking garage. …

Without knowing more about the distribution, it could be anything.

It's late, so I'll just do the case $\theta = 0$. Thus $X = (X_1,\ldots,X_n)^T$ has a multivariate normal distribution with mean $0$ and covariance matrix $I$. Let $U$ be an $n \times n$ orthogonal …

This is the way hypothesis testing works in classical statistics. Suppose e.g. you want to test whether factor $X$ affects $Y$. …

As soon as you see that the joint density factors as $f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3) = g(y_1,y_2) h(y_3)$ for some functions $g$ and $h$, it's clear that $(Y_1,Y_2)$ and $Y_3$ will be indepedent: integr …

The first step is to thoroughly understand the scenario that is being described, and what quantity the random variable represents. Each of the common distributions has a typical paradigm for what it …

Hint: try $X$ and $X^2$ where the distribution of $X$ is symmetric about $0$.

The second-order condition for a maximum of $G(x_1, \ldots, x_n)$ says that the Hessian matrix $$ H_{ij} = \dfrac{\partial^2 G}{\partial x_i \partial x_j}$$ is negative semidefinite. So for the case o …

Dirichlet showed that the number of pairs of positive integers $(a,b)$ with $ab \le x$ is $x \log x + (2 \gamma - 1)x + O(\sqrt{x})$ where $\gamma$ is Euler's constant (see e.g. http://www.math.uiuc. …

$\mathbb E(X) = \mu$ so $\mathbb E(X-\mu) = 0$. This is because: Expected value is linear. Expected value of a constant is that constant.

Let $F(k,n)$ be number of $k$-tuples such that $X_1 \le \ldots \le X_k$. Conditioning on $X_k$, we get $$F(k,n) = \sum_{m=1}^n F(k-1, m)$$ Thus $F(k,n) - F(k,n-1) = F(k-1,n)$. Boundary conditions are …

I presume the lengths of the sticks are supposed to be independent. Hint: the shortest stick has length $> x$ if and only if all $20$ sticks have length $> x$.

An outlier is a data point that is far from all other data points. So the only way it can change a mode is if all data points are unique, i.e. they are all in a tie for mode (in such a scenario it's …

You might try a least-squares fit to some simple form. Without any more information about the sorts of formulas that might be appropriate, I might try $c = p_0 + p_1 a + p_2 b$, which gives not too b …