# Category Archives: Expectations

## Minimum Dice Roll

Suppose that three fair 6-sided dice are rolled.

1. Let $$M$$ be the minimum of three numbers rolled. Find $$\mathbb{E}(M)$$.
2. Let $$S$$ be the sum of the largest two rolls. Find $$\mathbb{E}(S)$$.

## Dinner Party Seating

A host invites $$n$$ guests to a party (guest #1, guest #2, … , guest #n). Each guest brings with them their best friend. At the party there is a large circular table with \2n\) seats. All of the $$n$$ invited guests and their best friends sit in a random seat.

1. What is the probability that guest #1 is seated next to their best friend?
2. What is the expected number of the $$n$$ invited guests who are seated next to their best friend?

## A Dice Rolling Game

15 players each roll a fair 6-sided die once. If two or more players roll the same number, those players are eliminated. What is the expected number of players who get eliminated?

## January Birthdays at a Call Center

Calls arrive at a call center according to a Poisson arrival process with an average rate of 2 calls/minute. Each caller has a 1/12 chance of having a January birthday, independent of other callers. What is the expected wait time until the call center receives 3 calls from callers with January birthdays?

## Another Birthday Problem

A host invites guests to a party. How many guests should be invited in order for the expected number of guests who share a birthday with at least one other guest to be at least 4?

## Almost geometric

An experimenter rolls a fair 6-sided die until they’ve seen both a 1 and a 2 (not necessarily consecutively). What is the experimenter’s expected number of rolls?

## Handing back tests

A professor randomly hands back test in a class of $$n$$ people paying no attention to the names on the paper. Let $$N$$ denote the number of people who got the right test. Let $$D$$ denote the pairs of people who got each others tests. Let $$T$$ denote the number of groups of three who none got the right test but yet among the three of them that have each others tests. Find:

1. $$\mathbf{E} (N)$$
2. $$\mathbf{E} (D)$$
3. $$\mathbf{E} (T)$$

## Using the ﻿﻿Cauchy–Schwarz inequality

Recall that the Cauchy–Schwarz inequality stats that for any two random variable $$X$$ and $$Y$$ one has that
$\mathbf E |XY| \leq \sqrt{\mathbf E [X^2]}\,\sqrt{ \mathbf E [Y^2]}$

1. Use it to show that
$\mathbf E |X| \leq \sqrt{\mathbf E [X^2]}$

## A p.m.f. and expectation example

Let $$X$$ be a random variable with probability mass function

$p(n) = \frac{c}{n!}\quad \text{for \mathbf{N}=0,1,2\cdots}$

and $$p(x)=0$$ otherwise.

1. Find $$c$$. Hint use the Taylor series expansion of $$e^x$$.
2. Compute the probability that $$X$$ is even.
3. Computer the expected value of $$X$$

[Meester ex 2.7.14]

## Tail-sum formula for continuous random variable

Let $$X$$ be a positive random variable with c.d.f $$F$$.

1. Show using the representation $$X=F^{-1}(U)$$ where $$U$$ is $$\textrm{unif}(1,0)$$ that $$\mathbf{E}(X)$$ can be interpreted as the area above the graph on  $$y=F(x)$$ but below the line $$y=1$$. Using this deduce that
$\mathbf{E}(X)=\int_0^\infty [1-F(x)] dx = \int_0^\infty \mathbf{P}(X> x) dx \ .$
2. Deduce that if $$X$$ has possible values $$0,1,2,\dots$$ , then
$\mathbf{E}(X)=\sum_{k=1}^\infty \mathbf{P}(X\geq k)$

## probability density example

Suppose  $$X$$ takes values in$$(0,1)$$ and has a density

$f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\ 0 & x \not \in (0,1)\end{cases}$

for some $$c>0$$.

1. Find $$c$$.
2. Find $$\mathbf{E}(X)$$.
3. Find $$\mathrm{Var}(X)$$.

## Infinite Mean

Suppose that $$X$$ is a random variable whose density is

$f(x)=\frac{1}{2(1+|x|)^2} \quad x \in (-\infty,\infty)$

1. Draw a graph of $$f(x)$$.
2. Find $$\mathbf{P}(-1 <X<2)$$.
3. Find $$\mathbf{P}(X>1)$$.
4. Is $$\mathbf{E}(X)$$ defined ? Explain.

## Expectation of min of exponentials

There are $$15$$ stock brokers. The returns (in thousands of dollars) on each brokers is modeled as a separate independent exponential distribution $$X_1 \sim \mbox{Exp}(\lambda_1),…,X_{15} \sim \mbox{Exp}(\lambda_{15})$$. Define $$Z = \min\{X_1,…,X_{15}\}$$.

What is $$\mathbf{E}(Z)$$ ?

## Expectation of hierachical model

Consider the following hierarchical random variable

1. $$\lambda \sim \mbox{Geometric}(p)$$
2. $$Y \mid \lambda \sim \mbox{Poisson}(\lambda)$$
Compute $$\mathbf{E}(Y)$$.

## Expectation of geometric

Use the expectation as tail sum tool to compute the expectation of the geometric distribution.

## Expectation of mixture distribution

Consider the following mixture distribution.

1. Draw $$X \sim \mbox{Ber}(p=.3)$$
2. If $$X=1$$ then $$Y \sim \mbox{Geometric}(p_1)$$
3. If $$X= 0$$ then  $$Y \sim \mbox{Bin}(n,p_2)$$

What is $$\mathbf{E}(Y)$$ ?. (*) What is $$\mathbf{E}(Y | X )$$ ?.

## A simple mean calculation

Suppose that $$X \in \{1,2,3\}$$ and $$Y = X+ 1$$, and $$\mathbf{P}(X = 1) = 0.3, \ \mathbf{P}(X = 2) = 0.5,\ \mathbf{P}(X = 3) = 0.2.$$

(a) Find $$\mathbf{E}(X)$$.

(b) Find $$\mathbf{E}(Y)$$.

(c) Find $$\mathbf{E}(X + Y)$$.

[Author Mark Huber. Licensed under Creative Commons.]

## Expected Value and Mean Error

Let $$X$$ be a random variable with $$\mu_1=\mathbf{E}(X)$$ and $$\mu_2=\mathbf{E}(X^2)$$. For any number $$a$$ define the mean squared error

$J(a)=\mathbf{E}\big[(X-a)^2\big]$

and the absolute error

$K(a)=\mathbf{E}\big[|X-a|\big]$

1. Write $$J(a)$$ in terms of  $$a$$, $$\mu_1$$, and $$\mu_2$$ ?
2. Use the above answer to calculate $$\frac{d J(a)}{d\, a}$$ .
3. Find the $$a$$ which is the  solution to $$\frac{d J(a)}{d\, a}=0 ?$$ Comment on this answer in light of the name  “Expected Value” and argue that it is actually a minimum.
4. Assume that $$X$$ only takes values $$\{x_1,x_2,\dots,x_n\}$$.  Use the fact that
$\frac{d\ }{d a} |x-a| = \begin{cases} -1 & \text{if $$a < x$$}\\ 1 & \text{if $$a > x$$}\end{cases}$
to show that as long as $$a \not\in \{x_1,x_2,\dots,x_n\}$$ one has
$\frac{d K(a)}{d\, a} =\mathbf{P}(X<a) – \mathbf{P}(X>a)$
5. Now show that if $$a \in (x_k,x_{k+1})$$ then $$\mathbf{P}(X<a) – \mathbf{P}(X>a) = 2\mathbf{P}(X \leq x_k) – 1$$.
6. The median is any point $$a$$ so that both  $$\mathbf{P}(X\leq a) \geq \frac12$$ and $$\mathbf{P}(X\geq a) \geq\frac12$$. Give an example where the median is not unique. (That is to say there is more than one such $$a$$.
7. Use the above calculations  to show that if $$a$$ is any median (not equal to one of the $$x_k$$), then it solves  $$\frac{d K(a)}{d\, a} =0$$ and that it is a minimizer.

## Three Valued Random Variable

Show that the distribution of a random variable $$X$$ with possible values of 0,1  and 2 is determined by $$\mu_1=\mathbf{E}(X)$$ and $$\mu_2=\mathbf{E}(X^2)$$, by finding a formula for $$\mathbf{P}(X=x)$$ in terms of $$\mu_1$$ and $$\mu_2$$.

[Pitman p. 184. #20]

## Expectation of geometric distribution

Compute the expectation of the geometric distribution using the fact that in this case

$$\mathbf{E}(X)= \sum_{k=1}^{\infty} \mathbf{Pr}(X\geq k)$$

## Expectations of die rolls

A fair die is rolled ten times. Find numerical values for the expectations of each of the following random variables

1. the sum of the numbers in the ten rolls;
2. the sum of the largest two numbers in the first three rolls;
3. the maximum number in the first five rolls;
4. the number of multiples of three in the first ten rolls;
5. the number of faces which fail to appear in the ten rolls;
6. the number of different faces that appear in the ten rolls.

[From Pitman page 183]

## Subsequence problem

Scoring subsequences or lengths of similar matches or runs is common to a variety of problems from matches in genetic codes to similar runs in bits.

Consider the following question about two sequences of letters. Set both sequences to have length $$k$$. At each location of the sequences the probability of a match in letters is $$.7$$ and the probability of a mismatch is $$.3$$. At each location a match is assigned a score of $$4$$ and a mismatch is assigned a score of $$-1$$. The total score of the sequence is the sum of the scores at each location, there are $$k$$ locations.

1. What is the PMF of the total score if $$k=5$$.
2. What is the PMF of the total score for a general $$k$$ ?

## Putting expectations together

Suppose $$\mathbf{E}(X^2)=3$$, $$\mathbf{E}(Y^2)=4$$ and $$\mathbf{E}(XY)=2$$. What is  $$\mathbf{E}[(X+Y)^2]$$ ?

## Expection and dice rolls

A standard 6 sided die is rolled three times.

1. What is the expected value of the first roll ?
2. What is the expected values of the sum of the three rolls ?
3. What is the expected number of twos appearing in the three rolls ?
4. What is the expected number of sixes appearing in the three rolls ?
5. What is the expected number of odd numbers ?

Based on [Pitman, p. 182 #3]

## Indicatior functions and expectations

Let $$A$$ and $$B$$ be independent events and let $$\mathbf{1}_A$$ and $$\mathbf{1}_B$$ be the associated indicator functions. Answer the following questions in terms of $$\mathbf{P}(A)$$ and $$\mathbf{P}(B)$$.

1. Describe the distribution of $$\mathbf{1}_A$$.
2. What is $$\mathbf{E} \mathbf{1}_A$$ ?
3. Describe the distribution of $$(\mathbf{1}_A +\mathbf{1}_B)^2$$.
4. What is $$\mathbf{E}(\mathbf{1}_A +\mathbf{1}_B)^2$$ ?