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8/19 - Welcome to Prob 140 Fall 2020!

Prob 140 Math Prerequisites

A. Adhikari, Fall 2020

The course assumes fluency with all of the math below. The concepts, results, and methods will be used without explanation. If you’ve forgotten some of it, please review the relevant materialss. The more confident you are with this math content, the faster you will be able to grasp the material of the course.

Please do the exercises. They are based on calculations that come up frequently.

Being able to assess your own work is an important mathematical skill and will be crucial in this course. If your answer to an exercise below doesn’t match the one provided, or if you can’t get started on an exercise,

  • don’t ask for a detailed solution.
  • Remember that this is prereq material: you could do this stuff once upon a time. Maybe go back to your own previous class notes if you have them, or review the summaries here again, and examine your answer.
    • Which results did you use, and why?
    • Did you try a simpler version of the problem for which you can check your answer long-hand, for example by writing out all possibilities?
    • Did you draw a diagram?
    • Is your answer too big, or too small? Try to figure out which one, as it can help point out your error. If you can compare the question with another related one, that also can sometimes help explain the difference.
  • After going through the above, try the exercise again a couple of times.
  • If it still doesn’t work out, post on Piazza explaining the reasoning that you used and which of the above steps you took to troubleshoot.

If you have persistent trouble with these prereqs, you might want to try the course in a future term after strengthening your math, or try a different probability course this term. Or you have to be ready to spend a considerable amount of time learning the math and the probability simultaneously in this course this term. It’s possible but not easy.

Chapters 1-9

Basic Counting

The number of finite sequences and subsets that can be created out of a finite set of objects:

Review: Notation and Fundamental Properties If the math doesn’t render, please reload the page.

Notice the distinction between sequences (orderings, or “arrangements in a line”) and subsets (unordered collections). You have to decide which is appropriate, by looking carefully at the context. In some probability calculations it won’t matter which you decide to use, as long as you are consistent in the numerator and denominator of your proportions.

Exercise BC1: How many different ways are there to arrange six people

a) in a row?

b) in a circle?

Answer BC1 a) 720 b) 120

Exercise BC2: A committee consists of six women and four men. How many different choices can be made if you want to select

a) a Chairperson and an Assistant Chairperson?

b) a subcommittee of two people?

c) a subcommittee of three men and three women?

Answer BC2 a) 90 b) 45 c) 80

Sums

Summation notation will be used throughout the term. For Chapters 1-7 you just need the following.

Review: Terrific notes by Prof. James Aspnes of Yale’s CS department

“Sigma notation” is simply a compact way of making “…” precise when you are adding a bunch of terms. You’ve known the properties of sums since you started addition and multiplication in elementary school. The notation allows you to express those calculations concisely and precisely when the number of terms is large or when the terms being added are themselves functions.

  • Review the definition in the first sentence of the notes.
  • Then go to Section 1.4 which has standard notation that we will use.
  • You should understand (not just memorize) the first two sums in Section 2.1 on Page 5, and the first two identities in Section 2.2 on Page 6. There’s a typo in the second one. It should read

\(\sum_{i \in S} (x_i + y_i) ~ = ~ \sum_{i \in S} x_i + \sum_{i \in S} y_i\)

  • Section 2.3 has excellent advice.
  • In Chapter 4 of the textbook you’ll start working with double sums. The main idea is in Section 1.7 on Pages 4-5 of the Prof. Aspnes notes linked above; note the analogy with nested for loops. Useful properties are in the latter half of Section 2.2.

Exercise S1: Consider the sequence defined by $c_i =i$, for $i=1, 2, \ldots , 10$. If possible, find $\sum_{k=1}^{10} c_k$. If this is not possible, explain why not.

Answer S1 55

Exercise S2: Does the expression $\sum_{n=1}^{10} 2$ make sense? If it does, what is its value?

Answer S2 20

Exercise S3: Let ${c}$ and ${d}$ be sequences of real numbers so that \(\sum_{i=1}^{100} c_i ~=~ 10 ~~~~\mbox{and} ~~~~~ \sum_{j=1}^{100} d_j ~=~ 20\)

a) Find the value of $\sum_{i=1}^{100} (4c_i + 5)$.

b) Find the value of $\sum_{i=1}^{100} 4c_i ~+~ 5$.

c) Find the value of $\sum_{i=1}^{100} (4c_i - d_i + 5)$.

d) True or false: \(\sum_{i=1}^{100} \sum_{j=1}^{100} (c_i + d_j) ~~=~~ \sum_{i=1}^{100} (c_i + d_i)\) If the identity is true, find the common value of the two sides. If it is false, find the values of the two sides.

Answer S3 a) 540 b) 45 c) 520 d) False: left = 3000, right = 30

Exercise S4: Fill in the blanks:

\[\sum_{i=1}^n \sum_{j=i}^n a_{ij} ~ = ~ \sum_{j = \underline{~~~}}^{\underline{~~~}} \sum_{i = \underline{~~~}}^{\underline{~~~}} a_{ij}\]

[It might help to draw a grid of all $(i,j)$ pairs for some small value of $n$.]

Answer S4 $\sum_{j = 1}^n \sum_{i = 1}^j a_{ij}$

Induction

Mathematical induction is a method that is sometimes helpful for proving math statements if:

  • You have a guessed a math statement for each positive integer $n$, where the $n$th statement depends on $n$, and
  • for small $n$ (even just $n=1$) you can easily show that the statement is true.

Review: A concise and clear exposition from Prof. Victor Adamchik of Carnegie Mellon’s CS department

  • Read The Principle of Mathematical Induction on the first page. You don’t need the “more formal notation”.
  • Do the First Example under Induction Examples. It starts on the bottom of the second page.

The method is always the same.

  • Prove the base case.
  • Assume the induction hypothesis. That is, assume that the statement is true for a generic integer $n$.
  • Write the statement for $n+1$. This is what you have to prove.
  • [This is where the bulk of the work comes in.] Somehow write the statement for $n+1$ in terms of the statement for $n$ (assumed to be true) and the base case (which you have proved), in such a way that a little algebraic or other mathematical messing around show that the statement for $n+1$ is true.

Exercise I1: Show by induction that for each positive integer $n$, $\sum_{i=1}^n i ~ = ~ \frac{n(n+1)}{2}$.

Answer I1 The key step is $\sum_{i=1}^{n+1} i ~ = ~ \sum_{i=1}^n i ~+~ (n+1)$.

Exercise I2: Use I1 (no induction necessary) to find $\sum_{i=0}^{n-1} i$. This is the form in which the result first appears in the course, in Chapter 1 of the textbook.

Exercise I3: Use I1 and properties of sums (no induction necessary) to find a simple expression for the sum of the first $n$ odd integers: $\sum_{i=1}^n (2i-1)$.

Exercise I4: As a check, use induction to prove the formula you got in I3.

The Exponential and Log Functions

NOTE: All $\log$s in this class, and in almost all of math, are natural logarithms taken to the base $e$.

You know that $\log(1) = 0$. What we’re going to need, quite often, is an approximation for the log of a number that is very close to 1. A crude approximation is 0 because the number is close to 1. But we’ll use a finer approximation based on the first couple of terms in a related Taylor expansion.

Review: Graphs and Relevant Properties

For now, you need the Limits and Approximations section but not the Bounds. Preferably, you should understand the approximations in relation to the graphs of $e^x$ and $\log(x)$.

  • In Chapter 1 you’ll need the approximations.
  • Starting with Chapter 6 you’ll need the Taylor expansion of $e^x$.

Exercise EL1: About how big is

a) $\log(1.01)$?

b) $\log(0.99)$?

Answer EL1 a) 0.01 b) -0.01

Exercise EL2: Suppose $0 < p_n < 1$ for all $n$, $\lim_{n \to \infty} p_n = 0$, and $\lim_{n \to \infty} np_n = \mu$ for some number $\mu > 0$. Find $\lim_{n \to \infty} (1 - p_n)^n$.

[Hint: Start by considering the $\log$ of $(1 - p_n)^n$. Approximate it for large $n$, and then invert.

Answer EL2 $e^{-\mu}$

Exercise EL3: In EL2, $1 - p_n \to 1$ as $n \to \infty$, and $1^n = 1$ for all $n$. So why isn’t it true that $(1 - p_n)^n \to 1$ as $n \to \infty$?

Answer EL3 For a *fixed* power $m$, it's true that $(1 - p_n)^m \to 1$ as $n \to \infty$. But in the expression $(1 - p_n)^n$ the power $n$ isn't fixed. It's going to infinity. So the sequence $(1-p_n)^n$ is pulled in opposite directions: $1-p_n$ is heading for its upper limit of 1, but it's always less than 1 and so raising it to an increasing power $n$ keeps pulling it downwards.

Exercise EL4: Computers can’t do infinite sums (though they can get close numerically, and some symbolic math systems can handle many infinite sums). Find simple expressions for the following sums.

a) $ \sum_{n=0}^{\infty} \frac{1}{n!} $

b) $ \sum_{i=0}^{\infty} \frac{2^{3i}}{i!} $

c) $ \sum_{k=2}^{\infty} \frac{3^k}{k!} $

d) $\sum_{i=0}^\infty \frac{2^i}{(i+1)!}$

Answer EL4 a) $e$ b) $e^8$ c) $e^3 - 4$ d) $\frac{1}{2}(e^2 - 1)$

Geometric Series

We’ll use the infinite series more frequently than the finite one, starting in Chapter 8 of the textbook. In fact the infinite series is easier to sum (provided you assume it’s finite, which it’s fine for you to do), and then you can derive the finite series sum from the infinite one, as in the notes linked below.

Review: The main results on Page 6 (Section 2.1) of Prof. Aspnes notes. Understand the derivation of the infinite sum. That way you’ll never have to memorize the results.

Exercise GS1: Let $0 < p < 1$. Let $ S = \sum_{i=0}^{\infty} p^i $.

a) Find $S$.

b) Fill in the blank with the appropriate factor: $\sum_{i=3}^{\infty} p^i ~ = ~ \underline{~~~~~~~} \cdot S$. Hence find $ \sum_{i=3}^{\infty} p^i$.

c) Find $\sum_{i=0}^{\infty} p^{3i} $.

Answer GS1 a) $\frac{1}{1-p}$ b) The factor is $p^3$ so the sum is $\frac{p^3}{1-p}$. c) $\frac{1}{1-p^3}$

Chapters 10-25: Math Prereq Guide Under Construction