Problem set 1: Expectation of Continuous Random Varibles
(On the Moivre–Laplace theorem) 1800 dice are thrown. Find an approximate value for the probability that the total number of occurrences of 2 and 6 is not less than 620.
(On the Poisson limit theorem) When typing, a stenographer makes a mistake in a character with probability \(0.0005\). Find an approximate value for the probability that, when typing \(10{,}000\) characters, the stenographer will make a mistake no more than three times.
Find the expected value and variance of a random variable with the following distributions:
\(\xi \sim \mathcal N(a,\sigma)\).
\(\xi \sim \Gamma(\alpha,\beta)\).
\(\xi\) has a Beta distribution with parameters \((\alpha,\beta)\).
In a triangle with sides \(3\), \(4\), and \(5\), a random point \(X\) is selected. Let \(\xi\) be a random variable equal to the sum of the distances from \(X\) to each of the sides of the triangle. Find \(\mathbb E\xi\).
Let \(\xi_1,\xi_2,\ldots,\xi_n\) be independent random variables uniformly distributed on \([0,1]\). Find
\[ \mathbb E\xi_{(k)} \qquad\text{and}\qquad \operatorname{Var}\xi_{(k)}. \]
Let \(\xi\sim\mathcal N(0,\sigma^2)\) and \(k\in\mathbb N\). Find
\[ \mathbb E\xi^k \qquad\text{and}\qquad \mathbb E|\xi|^k. \]
Let \(\xi\) be a random variable with CDF
\[ F(x) = x\,\mathbf 1_{\left\{\frac14\le x\le \frac12\right\}} + \mathbf 1_{\left\{x\ge \frac12\right\}}. \]
Find \(\mathbb E\xi\) and \(\operatorname{Var}\xi\).
Do there exist random variables \(\xi\) and \(\eta\) such that both have the standard normal distribution and
\[ \operatorname{Cov}(\xi,\eta)=0, \]
but \(\xi\) and \(\eta\) are not independent?
The PDF of \(\eta\) is given by
\[ p_\eta(x) = \begin{cases} A\sin 2x, & x\in\left[0,\frac{\pi}{2}\right],\\ 0, & \text{otherwise}. \end{cases} \]
Find:
the value of \(A\);
the CDF of \(\eta\);
\(\mathbb E\eta\) and \(\operatorname{Var}\eta\);
\(\mathbb P\left(-\frac{\pi}{6}\le \eta\le \frac{\pi}{6}\right)\).
The coordinates of two points on the line are independent and uniformly distributed on \([0,1]\). Find the expected value and the variance of the distance between these two points.
A sniper at a shooting range shoots at the quarter-circle
\[ D=\left\{(x,y):x^2+y^2<1,\ x>0,\ y>0\right\}. \]
The random vector \((\xi,\eta)\) represents the hit point of the shooter and is uniformly distributed on \(D\). Find
\[ \operatorname{Cov}(\xi,\eta). \]
Let the random vector \((X,Y)\) have density
\[ p_{X,Y}(x,y) = \frac{1}{2\pi\sqrt{1-\rho^2}} \exp\left( -\frac{x^2-2\rho xy+y^2}{2(1-\rho^2)} \right). \]
Find the covariance matrix of \((X,Y)\) and the distribution of \(X\).
Find the expected value and variance of a random variable with the following distributions:
\(\xi\sim U[a,b]\).
\(\xi\sim \operatorname{Exp}(\lambda)\).
\(\xi\sim \operatorname{Laplace}\left(0,\frac1\lambda\right)\).
Consider random variables
\[ \xi\sim\mathcal N(0,1), \qquad \eta\sim\operatorname{Ber}\left(\frac12\right), \]
such that \(\xi-\eta\) and \(\eta\) are independent. Find
\[ \mathbb E e^{\xi+\eta}. \]
Let \(\xi\) have the following CDF:
\[ F_\xi(x) = \begin{cases} 0, & x<-2,\\[2mm] \frac15, & -2\le x<1,\\[2mm] \frac{x^2}{4}, & 1\le x<2,\\[2mm] 1, & x\ge 2. \end{cases} \]
Find \(\mathbb E\xi\) and \(\operatorname{Var}\xi\).
Let \(\xi_1,\xi_2,\ldots,\xi_n\) be i.i.d. random variables. Find
\[ \mathbb E\left[ \frac{\xi_1}{\xi_1+\cdots+\xi_n} \right]. \]
Let \(\xi_1\) and \(\xi_2\) be independent random variables with distribution \(\operatorname{Exp}(\lambda)\). Find
\[ \operatorname{Cov}\left( \frac{\xi_1}{\xi_1+\xi_2}, \,\xi_1+\xi_2 \right). \]
The joint PDF of a random vector \((\xi,\eta)\) is given by
\[ p_{\xi,\eta}(x,y) = \begin{cases} Axy, & (x,y)\in D,\\ 0, & (x,y)\notin D, \end{cases} \]
where
\[ D=\{(x,y):x\ge0,\ y\ge0,\ x+y\le2\}. \]
Find:
the value of \(A\);
\(\mathbb P((\xi,\eta)\in G)\), where
\[ G=\{(x,y):0\le x\le1,\ 0\le y\le1\}; \]
\(p_\xi(x)\) and \(p_\eta(y)\). Are \(\xi\) and \(\eta\) independent?
\(\mathbb E\xi\) and \(\mathbb E\eta\);
\(\operatorname{Var}\xi\) and \(\operatorname{Var}\eta\);
\(\operatorname{Cov}(\xi,\eta)\) and \(\operatorname{Corr}(\xi,\eta)\).