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Ordered Random Variables
Suppose \(X\) and \(Y\) are two random variable such that \(X \geq Y\).
- For a fixed number \(T\), which would be greater, \(\mathbf{P}(X \leq T) \) or \(\mathbf{P}(Y \leq T) \).
- What if \(T\) is a random variable ? (If it helps you think about the problem, assume \(T\) takes values in \(\{1,\cdots,n\}\). )
Coin tosses: independence and sums
A fair coin is tossed three times. Let \(X\) be the number of heads on the first two tosses, \(Y\) the number of heads on the last two tosses.
- Make a table showing the joint distribution of \(X\) and \(Y\).
- Are \(X\) and \(Y\) independent ?
- Find the distribution of \(X+Y\) ?
Poker Hands: counting
Assume that each of Poker hands are equally likely. The total number of hands is
\[\begin{pmatrix} 52 \\5\end{pmatrix}\]
Find the probability of being dealt each of the following:
- a straight flush ( all cards of the same suit and in order)
- a regular straight (but not a flush)
- two of a kind
- four of a kind
- two pairs (but not four of a kind)
- a full house (a pair and three of a kind)
In all cases, we mean exactly the hand stated. For example, four of a kind does not count as 2 pairs and a full house does not count as a pair or three of a kind.
Dice rolls: Explicit calculation of max/min
Let \(X_1\) and \(X_2\) be the number obtained on two rolls of a fair die. Let \(Y_1=\max(X_1,X_2)\) and \(Y_2=\min(X_1,X_2)\).
- Display the joint distribution tables for \( (X_1,X_2)\).
- Display the joint distribution tables for \( (Y_1,Y_2)\).
- Find the distribution of \(X_1X_2\).
Combination of [Pitman, p. 159 #4 and #5]
Blocks of Bernoulli Trials
In \(n+m\) independent Bernoulli \((p)\) trials, let \(S_n\) be the number of successes in the first \(n\) trials, \(T_n\) the number of successes in the last \(m\) trials.
- What is the distribution of \(S_n\) ? Why ?
- What is the distribution of \(T_m\) ? Why ?
- What is the distribution of \(S_n+T_m\) ? Why ?
- Are \(S_n\) and \(T_m\) independent ? Why ?
- Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
- Are \(S_{n+1}\) and \(T_{m}\) independent ? Why ?
Based on [Pitman, p. 159, #10]
The matching problem
There are \(n\) letters addressed to \(n\) eople at different addresses. The \(n\) addresses are typed on \(n\) envelopes. A disgruntled secretary shuffles the letters and puts them in the envelopes in random order, one letter per envelope.
- Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
- What is the probability approximately, for large \(n\) ?
[ For example, the needed inclusion-exclusion formula is given in Problem 12, p. 31 in Pitman]
You will also need to know the number of elements in the set
\[ \{ (i_1,i_2,\cdots, i_k) : 1 \leq i_1 < i_2 < \cdots < i_k\leq n\} \]
which is discussed here.
