Tag Archives: JCM_math230_HW4_S13

Ordered Random Variables

Suppose \(X\) and \(Y\) are two random variable such that \(X \geq Y\).

  1. For a fixed number \(T\), which would be greater, \(\mathbf{P}(X \leq T) \) or \(\mathbf{P}(Y \leq T) \).
  2. 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.

  1. Make a table showing the joint distribution of \(X\) and \(Y\).
  2. Are \(X\) and \(Y\)  independent ?
  3. 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:

  1. a straight flush ( all cards of the same suit and in order)
  2. a regular straight (but not a flush)
  3. two of a kind
  4. four of a kind
  5. two pairs (but not four of a kind)
  6. 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 \(Y_2=\min(X_1,X_2)\).

  1. Display the joint distribution tables for \( (X_1,X_2)\).
  2. Display the joint distribution tables for \( (Y_1,Y_2)\).
  3. 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.

  1. What is the distribution of \(S_n\) ? Why ?
  2. What is the distribution of \(T_m\) ? Why ?
  3. What is the distribution of \(S_n+T_m\) ? Why ?
  4. Are \(S_n\) and \(T_m\) independent ? Why ?
  5. Are \(S_n\) and \(T_{m+1}\) independent ? Why ?
  6. 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.

  1. Find the probability that at least one letter is put in a correctly addressed envelope. [Hint: use the inclusion-exclusion formula.]
  2. 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.