Expectation in the Simplest Cases

Let $\xi$ be the number of the web page chosen at random by a user from search results showing $M$ pages (the user chooses all pages with equal probability). Find $\mathbb {E}\left[\xi \right]$ and $\operatorname {Var}\left[\xi \right]$.

A spacecraft crew consisting of $k$ cosmonauts sets off to explore planets. The cosmonauts land at random on $m$ planets. The random variable $\xi$ equals the number of planets on which no one lands under this random placement. Find $E \xi$ and $D \xi$, if

1. the planets are indistinguishable;

2. the planets are distinguishable.

The random graph model $G(n, p)$ is considered. Find $\mathbb {E}\left[X\right]$, if

1. $X$ is the number of triangles (cycles of length 3) in the random graph;

2. $X$ is the number of cycles of length $k$ in the random graph;

3. $X$ is the number of cliques (subgraphs that are complete graphs) of size $k$ in the random graph.

Consider the random graph model $G(n, p)$. Find $\operatorname {Var}\left[X\right]$ if

1. $X$ is the number of triangles (cycles of length 3) in the random graph;

2. $X$ is the number of cliques (subgraphs that are complete graphs) of size $k$ in the random graph.

A user entered a search query 10 times. It is assumed that the time intervals between the $i$-th and $(i+1)$-th queries equal $\xi_{i}$ minutes, $i \in \left\{ 1, \ldots , 9\right\}$, where $\xi_{1}, \ldots , \xi_{9}$ are independent random variables, exponentially distributed with parameter 1. Let $X$ be the number of queries entered during the first 5 minutes. Find $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.

Given a random variable $\xi$. Find the expectation and variance of $\xi$ if it has

1. a binomial distribution with parameters $(n, p)$;

2. a geometric distribution with parameter $p$ (i.e. $\mathbb {P}\left(\xi =k\right)=p(1-p)^{k-1}, k=1,2, \ldots$);

3. a normal distribution with parameters $\left(a, \sigma^{2}\right)$;

4. a uniform distribution on the interval $[a, b]$;

5. a gamma distribution with parameters $(\alpha , \lambda )$;

6. a beta distribution with parameters $(\alpha , \beta )$.

A random variable $\xi$ has the standard normal distribution. Compute $\mathbb {E}\left[\xi^{k}\right]$ and $\mathbb {E}\left[\left|\xi \right|^{k}\right]$ for $k \in \mathbb {N}$. Compute the same characteristics if $\xi \sim \mathcal{N}\left(0, \sigma^{2}\right)$.

In a random graph $G(n, p)$, under certain conditions the number of occurrences of a fixed subgraph has a distribution close to Poisson. In proving this fact, one computes factorial moments of random variables (the $k$-th factorial moment of a random variable $\xi$ is defined as $\mathbb {E}\left[\xi (\xi -1) \ldots (\xi -k+1)\right]$). Let $\xi \sim \operatorname {Pois}(\lambda )$. Find the $k$-th factorial moment of the random variable $\xi$.