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Starting in on Chapter 2

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<title>Random Variables</title>
<introduction>
<p> Text before the first section. </p>
<p>
Text before the first section.
</p>
</introduction>
<section xml:id="sec-Discrete-RVs" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Discrete Random Variables</title>
<title>Discrete Random Variables</title>
<p> Text of section. </p>
<definition xml:id="def-RV">
<statement>
<p>
Let <m>\Omega</m> be a sample space.
A <term>random variable</term> is a function <m>X \colon \Omega \to \mathbb{R}</m> which assigns a number to each outcome.
</p>
</statement>
</definition>
<example>
<statement>
<p>
Suppose an experiment consists of flipping a fair coin 4 times.
Let <m>X</m> be the number of heads.
The actual outcomes in the experiment are heads/tails sequences of length 4, such as <m>HTHT</m> and <m>HHHT</m>.
The random variable <m>X</m> assigns a numerical measurement to the outcomes, such as <m>X(HTHT) = 2</m> and <m>X(HHHT) = 3</m>.
</p>
<p>
For any real number value <m>a</m>, we can consider the event consisting of outcomes such that <m>X = a</m>.
For example, the flip sequences <m>HHHT, HHTH, HTHH</m>, and <m>THHH</m> all have 3 heads, so <m>\Pr(X = 3) = \frac{4}{16} = \frac{1}{4}</m>.
</p>
</statement>
</example>
<example>
<title>Indicator Random Variable</title>
<statement>
<p>
Let <m>A</m> be an event in the sample space <m>\Omega</m>.
An <term>indicator random variable</term> for <m>A</m> is the random variable <m>X</m> such that
<md>
<mrow> X(\omega) = \begin{cases} 1 \amp \omega \in A \\ 0 \amp \omega \notin A \end{cases} </mrow>
</md>
</p>
</statement>
</example>
<p>
Indicator random variables are also called <term>Bernoulli random variables</term>, although we'll prefer the former term since it more clearly states the purpose of these random variables: to indicate whether or not a particular event has occurred.
We'll use the language "<m>X</m> indicates <m>A</m>" to mean that <m>X</m> is an indicator random variable for the event <m>A</m>.
In this case, <m>\Pr(X = 1) = \Pr(A)</m>, and <m>\Pr(X = 0) = 1 - \Pr(A)</m>.
</p>
<definition xml:id="def-discrete-and-continuous">
<statement>
<p>
Consider the set <m>S</m> of all real numbers which are actually output by a random variable <m>X</m>.
If <m>S</m> does not contain any interval of values, then the random variable <m>X</m> is called <term>discrete</term>.
Otherwise, it's called <term>continuous</term>.
</p>
</statement>
</definition>
<example>
<statement>
<p>
Any random variable <m>X</m> defined on a finite sample space <m>\Omega</m> is discrete<mdash/>the set of outputs of <m>X</m> cannot contain an interval if it only has finitely many values.
</p>
<p>
Suppose an experiment consists of growing a plant in a new fertilizer, and a random variable <m>Y</m> measures the height of the plant after a set growing period.
Now <m>Y</m> could conceivably take on an interval's worth of values (such as, e.g., any real number between 10 inches and 20 inches), so this random variable would be continuous.
</p>
<p>
The distinction between discrete and continuous random variables is not simply the distinction of whether the sample space is finite or infinite.
Consider the experiment in which we flip a coin repeatedly until we first see a coin come up heads.
Let <m>Z</m> be the number of times the coin is flipped. Then there are infinitely many possible values of <m>Z</m> (1, 2, 3, and so on), but there's no interval of real numbers which are all possible outputs of <m>Z</m>.
The possible outputs of <m>Z</m> are discrete (in the non-technical, English sense of the word: separated).
</p>
</statement>
</example>
<exercises xml:id="exercises-Discrete-RVs">
<exercise>
<statement>
<p>
We flip a coin with unknown bias 100 times and observe 43 heads.
What is the maximum likelihood estimation for the probability of the coin coming up heads?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
An experiment consists of flipping a biased coin 20 times.
If the coin comes up heads with probability <m>\theta = 0.3</m>, find the probability of seeing 5 heads.
Find the probability of seeing up to (and including) 3 heads.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
An experiment consists of flipping a coin repeatedly until we first see heads.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
If the coin comes up heads with probability 0.4, what is the probability we’ll see our first heads within three flips? What about precisely on the third flip?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
Suppose the coin has an unknown probability of coming up heads.
We perform the experiment in five independent trials, during which it takes 4, 5, 4, 3, and 6 flips to see our first heads in each trial.
What is the maximum likelihood estimation for the probability of the coin coming up heads on a flip?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
A particular store has an average of 20 customers each hour.
During a 4-hour afternoon shift, what is the probability of serving 80 customers.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
A particular store owner wants to approximate the average hourly rate at which customers come into the store.
They observe 80 customers enter during a particular 4-hour shift.
What is the maximum likelihood estimation for the hourly customer rate?
</p>
</statement>
</exercise>
</exercises>
</section>
<section xml:id="sec-Continuous-RVs" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Continuous Random Variables</title>
<title>Continuous Random Variables</title>
<p> Text of section. </p>
<p>
Text of section.
</p>
<exercises xml:id="exercises-Continuous-RVs">
<exercise>
<statement>
<p>
A continuous random variable <m>X</m> taking values in <m>[1, 4]</m> has p.d.f.
<m>f(x) = k(x - \sqrt{x})</m> for some constant <m>k</m>.
What is the value of <m>k</m>?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
A continuous random variable <m>X</m> taking values in <m>[1, 2]</m> has p.d.f.
<m>\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}</m>.
Find the c.d.f.
<m>F(x)</m>.
Use your c.d.f.
to find <m>\Pr\left(1 \leq X \leq \frac{3}{2}\right)</m>.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
A continuous random variable <m>X</m> taking values in <m>[2, 3]</m> has c.d.f.
<m>F(x) = \frac{x^3}{3} - x^2 + 4</m>.
Find the p.d.f.
<m>f(x)</m>.
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
A radioactive material emits particles at an unknown probabilistic rate <m>\lambda</m> particles per minute.
We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes.
Write the likelihood function <m>\mathcal{L}(\lambda)</m> based on this data.
What is the maximum likelihood estimation for <m>\lambda</m>?
</p>
</statement>
</exercise>
<exercise>
<statement>
<p>
Suppose a parameter <m>\theta</m> takes values in <m>[0, 1]</m> with likelihood function <m>\mathcal{L}(\theta) = \sqrt{\theta} - \theta^2</m>.
Find the maximum likelihood estimation of <m>\theta</m>.
</p>
</statement>
</exercise>
</exercises>
</section>
</chapter>