# 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.