Problem set 5: Expectation and Variance of Discrete Random Variables
During the seminars, we will work on problems of three different difficulty levels. Category 1 (green) contains introductory problems, Category 2 (blue) contains standard problems, and Category 3 (orange) contains the most challenging problems.
1 Category 1
A number is chosen at random from the set \(S = \{-1,0,1\}\). Let \(X\) be the number chosen.
Find the expected value, variance, and standard deviation of \(X\).
The random variable \(X\) takes the values \(0\), \(1\), and \(4\) according to the following probability distribution:
\[ \begin{array}{c|ccc} x & 0 & 1 & 4\\ \hline P_X(x) & 0.2 & k & k \end{array} \]
Determine the constant \(k\).
Find \(\mathbb E(X)\).
Find \(\operatorname{Var}(X)\).
A random variable \(X\) has the following distribution:
\[ \begin{array}{c|cccc} x & 0 & 1 & 2 & 4\\ \hline P_X(x) & \frac13 & \frac13 & \frac16 & \frac16 \end{array} \]
Find \(\mathbb E(X)\), \(\mathbb E(X(X+1))\), \(\operatorname{Var}(X)\), and \(\sigma(X)\).
\(X\) is a random variable with \(\mathbb E(X)=100\) and \(\operatorname{Var}(X)=15\). Find:
\(\mathbb E(X^2)\).
\(\mathbb E(3X+10)\).
\(\mathbb E(-X)\).
\(\operatorname{Var}(-X)\).
\(\sigma(-X)\).
A fair coin is tossed three times. Let \(X\) be the number of heads that turn up. Find \(\operatorname{Var}(X)\) and \(\sigma(X)\).
Let \(\xi_1,\xi_2\) represent the values obtained on two different dice. Find
\[ \mathbb E(\xi_1),\qquad \mathbb E(\xi_1+\xi_2),\qquad \mathbb E(\xi_1\xi_2). \]
2 Category 2
A random variable \(X\) has a binomial distribution with mean \(10\) and variance \(6\). Find
\[ P(X=4). \]
A box contains \(10\) white balls and \(2\) black balls. Six balls are selected at random. Random variable \(X\) is equal to the number of black balls among the selected balls.
Find the distribution of \(X\).
Find \(\mathbb E(X)\).
Find \(\mathbb E(X\mid X>0)\).
Find the expected value and the variance of a random variable \(\xi\) in each of the following cases:
\(\xi\sim \operatorname{Ber}(p)\).
\(\xi\sim \operatorname{Bin}(n,p)\).
\(\xi\sim \operatorname{Pois}(\lambda)\).
\(\xi\sim U\{1,\ldots,N\}\).
\(\xi\sim \operatorname{Geom}(p)\).
A box contains \(m\) white balls and \(n\) black balls. We randomly pick a ball, return it to the box, and continue until we obtain a white ball. Find the expectation and the variance of the number of balls taken out.
A group of \(n\) students throw their hats at their graduation. After that, each of them picks up from the floor a hat at random. Find the expected value and the variance of the number of students who get the correct hat.
Let \(\xi\sim \operatorname{Pois}(\lambda)\). Find
\[ \mathbb E e^{\xi}. \]
A basket contains \(4\) apples and \(6\) oranges. We randomly choose \(3\) fruits from it. Let \(\xi\) denote the number of apples among the selected fruits. Find
\[ \mathbb E\xi,\qquad \operatorname{Var}\xi,\qquad P(\xi>1). \]
Suppose we can roll a fair \(N\)-sided die up to \(m\) times. Each face of the die contains a distinct number of points from \(1\) to \(N\).
Let \(\xi\) correspond to the number of times that the face showing \(1\) point appears. In the same way, let \(\eta\) represent the sum of all the points obtained during the game.
Find
\[ \mathbb E\xi,\quad \operatorname{Var}\xi, \qquad \mathbb E\eta,\quad \operatorname{Var}\eta. \]
3 Category 3
For a natural number \(k\), the \(k\)-th factorial moment of a random variable \(\xi\) is
\[ \mathbb E[\xi(\xi-1)(\xi-2)\cdots(\xi-k+1)]. \]
Prove that if \(\xi\sim \operatorname{Pois}(\lambda)\), then its \(k\)-th factorial moment is
\[ \lambda^k. \]
A coin is tossed \(1000\) times. Using Chebyshev’s inequality, estimate the probability that the number of times we obtain tails will be in the interval
\[ [450,550]. \]
A person who has \(n\) keys wants to unlock his door by testing the keys independently and in random order. Keys that do not match are not excluded from further tests. It is known that only one of the available keys fits the lock. Find the expectation and variance of the number of trials.
- Solve the same problem under the assumption that keys that do not fit are excluded from further tests.