Problem set 4: 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 box contains two gold balls and three silver balls. You are allowed to choose balls successively from the box at random. You win 1 dollar each time you draw a gold ball and lose 1 dollar each time you draw a silver ball. After a draw, the ball is not replaced.
Show that, if you draw until you are ahead by 1 dollar or until there are no more gold balls, this is a favorable game.
Sketch the CDF for a random variable \(X\) with PMF
\[ \begin{array}{c|cccc} x & 0 & 1 & 2 & 4 \\ \hline P_X(x) & \frac13 & \frac13 & \frac16 & \frac16 \end{array} \]
Consider two random variables with the following joint distribution:
\[ \begin{array}{c|cc} X\backslash Y & 1 & 2 \\ \hline 3 & \frac14 & \frac14 \\ 5 & \frac16 & \frac13 \end{array} \]
Find the marginal distributions of \(X\) and of \(Y\).
Are \(X\) and \(Y\) independent?
A coin is tossed 3 times. Let the random variable \(\xi\) denote the number of tails obtained. Let \(\eta\) denote the winnings earned in a single play of a game with the following rules:
- The player wins $1 if the first tails occurs on the first toss.
- The player wins $2 if the first tails occurs on the second toss.
- The player wins $3 if the first tails occurs on the third toss.
- The player loses $1 if no tails occur.
Find the joint PMF of \(\xi\) and \(\eta\).
There are 100 cards in a box, labeled with numbers \(1,2,\ldots,100\). A card is drawn at random from the box and immediately returned to the box. We repeat this procedure 200 times.
Find the approximate value of the probability that the card with the number 1 will appear exactly 3 times.
There is a set of four cards marked with the numbers \(-1,0,1,2\). The player draws one card at random, writes down the number on the card, and returns the card back. Then the player repeats the procedure again.
Let \(X_i\) be the number written during the \(i\)-th experiment, \(i=1,2\).
Find the PMF of the random variables
\[ Y=X_1X_2 \]
and
\[ Z=X_1+X_2. \]
The joint PMF of the random vector \((X,Y)\) is given by the following table:
\[ \begin{array}{c|ccc} X\backslash Y & -1 & 0 & 1 \\ \hline -2 & \frac18 & \frac14 & \frac18 \\ 2 & \frac1{12} & \frac13 & \frac1{12} \end{array} \]
Find the PMF of \(X\) and \(P(Y\geq0)\).
Find the PMF of \(X+Y\).
Find the PMF of \(Z=\min\{X,Y\}\).
There is a set of four cards marked with the numbers \(-1,0,1,2\). The player draws one card at random, writes down the number on the card, and does not return the card back. Then the player repeats the procedure again.
Let \(X_i\) be the number written during the \(i\)-th experiment, \(i=1,2\).
Find the PMF of the random variables
\[ Y=X_1X_2 \]
and
\[ Z=X_1+X_2. \]
You roll one red die and one green die. Define the random variables \(\xi\) and \(\eta\) as follows:
\[ \xi = \text{the number showing on the red die}, \]
and
\[ \eta = \text{the number of dice that show the number six}. \]
For example, if the red and green dice show the numbers 6 and 4, then \(\xi=6\) and \(\eta=1\).
Write down the table showing the joint probability mass function for \(\xi\) and \(\eta\), find the marginal distribution for \(\eta\), and compute \(E\eta\).
The joint PMF of the random vector \((X,Y)\) is given by
\[ P(X=k,Y=j)=p^k(1-p)\frac{\lambda^j}{j!}e^{-j}, \qquad k=0,1,2,\ldots,\quad j=0,1,2,\ldots, \]
where \(0<p<1\) and \(\lambda>0\).
Find the marginal PMF of \(X\) and \(Y\).
2 Category 2
The PMF of \(\xi\) is given by
\[ P(\xi=k)=\frac{c}{k(k+1)(k+2)}, \qquad k=1,2,\ldots. \]
Find:
The value of \(c\).
\(P(\xi\geq3)\).
Let \(\xi\) and \(\eta\) be two independent random variables such that
\[ \xi\sim \mathrm{Bin}(n,p) \]
and
\[ \eta\sim \mathrm{Bin}(m,p). \]
Find the probability distribution of \(\xi+\eta\).
Let \(P\) be a discrete distribution on \(\mathbb R\), and let
\[ p(x)=P(\{x\}). \]
Suppose
\[ p(x)=\frac{\lambda^x e^{-\lambda}}{x!}, \qquad x\in\mathbb Z_+, \]
that is, \(P\) is a Poisson distribution.
Find
\[ P(2\mathbb Z_+). \]
A die is rolled \(k\) times. Let \(X_i\), \(i=1,2,\ldots,k\), be the number appearing on the upper face of the die during the \(i\)-th roll.
- Find the PMF of
\[ Y_k=\max\{X_1,\ldots,X_k\}. \]
- Find the PMF of
\[ Z_k=\min\{X_1,\ldots,X_k\}. \]
- Find the PMF of
\[ Z_k=\max\{4,X_1\}. \]
Let \(\xi\) and \(\eta\) be two independent random variables such that
\[ \xi\sim\mathrm{Poiss}(\lambda_1) \]
and
\[ \eta\sim\mathrm{Poiss}(\lambda_2). \]
Find the probability distribution of \(\xi+\eta\).
3 Category 3
Consider a sequence of independent random variables \(\{\xi_i\}_{i=1}^n\) such that
\[ \xi_i\sim\mathrm{Geom}(p_i). \]
Prove that
\[ \min\{\xi_1,\ldots,\xi_n\} \sim \mathrm{Geom}\left( 1-\prod_{i=1}^n(1-p_i) \right). \]
Let \(\xi\) and \(\eta\) be two independent random variables such that
\[ \xi\sim\mathrm{Poiss}(\lambda_1) \]
and
\[ \eta\sim\mathrm{Poiss}(\lambda_2). \]
Find
\[ P\{\xi=k\mid \xi+\eta=n\}. \]
Let \(P\) be a discrete distribution on \(\mathbb R\), and let
\[ p(x)=P(\{x\}). \]
Suppose
\[ p(x)=p(1-p)^{x-1}, \qquad x\in\mathbb N, \]
that is, \(P\) is a geometric distribution.
Find its CDF and
\[ P(2\mathbb Z_+). \]