From c7f39b769308edee327bb395b7a896bf04539407 Mon Sep 17 00:00:00 2001 From: andyeisenberg <125768484+andyeisenberg@users.noreply.github.com> Date: Wed, 31 Dec 2025 03:26:25 +0000 Subject: [PATCH] Starting in on Chapter 2 --- source/ch-Random-Variables.ptx | 212 +++++++++++++++++++++++++++++++-- 1 file changed, 202 insertions(+), 10 deletions(-) diff --git a/source/ch-Random-Variables.ptx b/source/ch-Random-Variables.ptx index 26bb724..3245914 100644 --- a/source/ch-Random-Variables.ptx +++ b/source/ch-Random-Variables.ptx @@ -4,25 +4,217 @@ Random Variables -

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+

+ Text before the first section. +

- Discrete Random Variables + Discrete Random Variables -

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+ + +

+ Let \Omega be a sample space. + A random variable is a function X \colon \Omega \to \mathbb{R} which assigns a number to each outcome. +

+ + + + +

+ Suppose an experiment consists of flipping a fair coin 4 times. + Let X be the number of heads. + The actual outcomes in the experiment are heads/tails sequences of length 4, such as HTHT and HHHT. + The random variable X assigns a numerical measurement to the outcomes, such as X(HTHT) = 2 and X(HHHT) = 3. +

+ +

+ For any real number value a, we can consider the event consisting of outcomes such that X = a. + For example, the flip sequences HHHT, HHTH, HTHH, and THHH all have 3 heads, so \Pr(X = 3) = \frac{4}{16} = \frac{1}{4}. +

+ + + + + Indicator Random Variable + + +

+ Let A be an event in the sample space \Omega. + An indicator random variable for A is the random variable X such that + + X(\omega) = \begin{cases} 1 \amp \omega \in A \\ 0 \amp \omega \notin A \end{cases} + +

+ + + +

+ Indicator random variables are also called Bernoulli random variables, 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 "X indicates A" to mean that X is an indicator random variable for the event A. + In this case, \Pr(X = 1) = \Pr(A), and \Pr(X = 0) = 1 - \Pr(A). +

+ + + +

+ Consider the set S of all real numbers which are actually output by a random variable X. + If S does not contain any interval of values, then the random variable X is called discrete. + Otherwise, it's called continuous. +

+ + + + + +

+ Any random variable X defined on a finite sample space \Omega is discretethe set of outputs of X cannot contain an interval if it only has finitely many values. +

+ +

+ Suppose an experiment consists of growing a plant in a new fertilizer, and a random variable Y measures the height of the plant after a set growing period. + Now Y 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. +

+ +

+ 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 Z be the number of times the coin is flipped. Then there are infinitely many possible values of Z (1, 2, 3, and so on), but there's no interval of real numbers which are all possible outputs of Z. + The possible outputs of Z are discrete (in the non-technical, English sense of the word: separated). +

+ + + + + + +

+ 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? +

+ + + + + +

+ An experiment consists of flipping a biased coin 20 times. + If the coin comes up heads with probability \theta = 0.3, find the probability of seeing 5 heads. + Find the probability of seeing up to (and including) 3 heads. +

+ + + + + +

+ An experiment consists of flipping a coin repeatedly until we first see heads. +

+ + + + + +

+ 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? +

+ + + + + +

+ 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? +

+ + + + + +

+ 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. +

+ + + + + +

+ 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? +

+ + +
- -
- Continuous Random Variables + Continuous Random Variables -

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+

+ Text of section. +

+ + + +

+ A continuous random variable X taking values in [1, 4] has p.d.f. + f(x) = k(x - \sqrt{x}) for some constant k. + What is the value of k? +

+ + + + + +

+ A continuous random variable X taking values in [1, 2] has p.d.f. + \displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}. + Find the c.d.f. + F(x). + Use your c.d.f. + to find \Pr\left(1 \leq X \leq \frac{3}{2}\right). +

+ + + + + +

+ A continuous random variable X taking values in [2, 3] has c.d.f. + F(x) = \frac{x^3}{3} - x^2 + 4. + Find the p.d.f. + f(x). +

+ + + + + +

+ A radioactive material emits particles at an unknown probabilistic rate \lambda 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 \mathcal{L}(\lambda) based on this data. + What is the maximum likelihood estimation for \lambda? +

+ + + + + +

+ Suppose a parameter \theta takes values in [0, 1] with likelihood function \mathcal{L}(\theta) = \sqrt{\theta} - \theta^2. + Find the maximum likelihood estimation of \theta. +

+ + +
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