auto-update(nvim): 2025-03-01 20:23:15

documents/by-course/pstat-120a/course-notes/main.typ

@@ -17,24 +17,8 @@ scribe's, not the instructor's.
 
 = Lecture #datetime(day: 6, month: 1, year: 2025).display()
 
-== Preliminaries
-
-#definition[
-  Statistics is the science dealing with the collection, summarization,
-  analysis, and interpretation of data.
-]
-
 == Set theory for dummies
 
-A terse introduction to elementary naive set theory and the basic operations
-upon them.
-
-#remark[
-  Keep in mind that without $cal(Z F C)$ or another model of set theory that
-  resolves fundamental issues, our set theory is subject to paradoxes like
-  Russell's. Whoops, the universe doesn't exist.
-]
-
 #definition[
   A *set* is a collection of elements.
 ]
@@ -2186,7 +2170,76 @@ indicator of where the center of the distribution lies.
 
 = President's Day lecture
 
-...
+== Quantiles
+
+#definition[
+  For $p in (0,1)$, the *$p^"th"$ quantile* of a random variable $X$ is any $x in RR$ satisfying
+  $
+    P(X >= x) >= 1 - p "and" P(X <= x) >= p
+  $
+]
+
+We see that the median is the $0.5^"th"$ quantile. $p = 0.25$ is called the
+"first quartile" (Q1). $p = 0.75$ is called the "third quartile" (Q3).
+
+$Q 3 - Q 1$ is called the $I Q R$, the interquartile range.
+
+== Variance
+
+Variance is a measure of spread or _variation_ from the mean. Variance is the
+*expected squared deviations* about the mean.
+
+#definition[
+  Let $X$ be a random variable with mean $mu$. The variance of $X$ is given by
+  $
+    "Var"(X) = E[(X-mu)^2] = sigma_X^2
+  $
+  If $X$ is discrete with PMF $p_X(x)$, then the variance is
+  $
+    "Var"(X) = sum_x (x-mu)^2 p_X (x)
+  $
+  If $X$ is continuous with PMF $f_X (x)$, then the variance is
+  $
+    "Var"(X) = integral^infinity_(-infinity) (x-mu)^2 f_X (x) dif x
+  $
+]
+
+Variance is the same as the second central moment.
+
+#fact[
+  $sigma_X = sqrt("Var"(X))$ is the "standard deviation" of $X$.
+]
+
+These tell us about how far spread out the points are.
+
+#example[Fair die][
+  Find the variance for the value of a single roll of a fair die.
+  $
+    sigma_X^2 = "Var"(X) &= E[(X-3.5)^2] \
+    &= sum_("all" x) (x-3.5)^2 dot p_X (x) \
+    &=91 / 6
+  $
+]
+
+#example[Continuous $X$][
+  Let $X$ be a continuous RV with PDF $f_X (x) = cases(1 &"for" 0 < x < 1, 0 &"otherwise")$
+
+  Find $E[X]$:
+  $
+    integral_0^1 x dot f_X (x) dif x = 1 / 2
+  $
+
+  Find $"Var"(X)$:
+  $
+    E[(X- 1 / 2)^2] &= integral_0^1 (x- 1 / 2)^2 dot f_X (x) dif x \
+    &= 1 / 12
+  $
+]
+
+An easier formulation of variance is given by
+$
+  "Var"(X) equiv E[(X-mu)^2] = E[X^2] - mu^2
+$
 
 = Lecture #datetime(day: 19, month: 2, year: 2025).display()
 
@@ -2462,3 +2515,68 @@ $
     p_(X_1,X_2,X_n) (k_1,k_2,...,k_n) >= 0
   $
 ]
+
+= Joint distributions
+
+== Introduction
+
+Looking at 2 or more random variables at the same time. Treat $n$ random
+variables as the coordinates of an $n$ dimensional *random vector*. In fact, like how a random variable is a function from $Omega -> RR$, the joint random vector is a vector-valued function
+$
+  vec(x,y) : Omega -> RR^2
+$
+The probability distribution of $(X_1,X_2,...,X_n)$ is now represented by
+$
+  P((X_1,X_2,...,X_n) in B)
+$
+where $B$ are subsets of $RR^n$. The probability distribution of the random
+vector is the *joint distribution*. The probability distribution of individual
+coordinates $X_j$ are *marginal distributions*.
+
+== Discrete joint distributions
+
+Let $X$ and $Y$ both be discrete random variables defined on a common $Omega$. Then the joint PMF is given by
+$
+  P(X=x, Y=y) equiv p_(X,Y) (x,y)
+$
+with the property that
+$
+  sum_("all" x) sum_("all" y) p_(X,Y) (x,y) = 1
+$
+
+#definition[
+  Let $X_1,X_2,...,X_n$ be discrete random variables defined on a common $Omega$, then their *joint probability mass function* is given by:
+  $
+    p(k_1,k_2,...,k_n) = P(X_1 = k_1, X_2 = k_2, ..., X_n = k_n)
+  $
+  for all possible values $k_1,k_2,...,k_n$ of $X_1,X_2,...,X_n$.
+]
+
+#fact[
+  The joint probability in set notation looks like
+  $
+    P(X_1 = k_1, X_2 = k_2, ..., X_n = k_n) = P({X_1=k_1} sect {X_2 = k_2} sect dots.c sect {X_n=k_n})
+  $
+  The joint PDF has the same properties as the PDF for the single random variable, namely
+  $
+    p_(X_1,X_2,...,X_n) (k_1,k_2,...,k_n) >= 0 \
+    sum_(k_1,k_2,...,k_n) p_(X_1,X_2,...,X_n) (k_1,k_2,...,k_n) = 1
+  $
+]
+
+#fact[
+  Let $g : RR^n -> RR$ be a real-valued function on an $n$-vector. If $X_1,X_2,...,X_n$ are discrete random variables with joint PMF $p$ then
+  $
+    E[g(X_1,X_2,...,X_n)] = sum_(k_1,k_2,...,k_n) g(k_1,k_2,...,k_n) p(k_1,k_2,...,k_n)
+  $
+  provided the sum is well defined.
+]
+
+#example[
+  Flip a fair coin three times. Let $X$ be the number of tails in the first flip and $Y$ a total number of tails observed from all flips. Then the support of each variable is $S_X = {0,1}$ and $S_Y = {0,1,2,3}$.
+
+  1. Find the joint PMF of $(X,Y)$, $p_(X,Y) (x,y)$.
+    Just record the probability of the respective events.
+
+    For example, the probability $X$ is 0 and $Y$ is 1 is $p_(X,Y) (0,1)$ is $2/8$.
+]