Merge branch 'main' of https://git.47c.in/arc/notes

.obsidian/plugins/obsidian-git/data.json

@@ -2,7 +2,7 @@
   "commitMessage": "vault backup: {{date}}",
   "autoCommitMessage": "vault backup: {{date}}",
   "commitDateFormat": "YYYY-MM-DD HH:mm:ss",
-  "autoSaveInterval": 1,
+  "autoSaveInterval": 5,
   "autoPushInterval": 0,
   "autoPullInterval": 5,
   "autoPullOnBoot": false,

education/math/Inverse Functions.md

@@ -0,0 +1,10 @@
+For a function to have an inverse, it needs to have one $x$ for every $y$, and vice versa. You can use the horizontal line test to verify that the inverse of a function is valid. If you can draw a horizontal line and it crosses through two points at the same time at any height, the inverse is not a valid function. To get the inverse, you can switch the x and y of a function, and it will mirror the graph over the line $y = x$. 
+
+# Examples
+Given the below function:
+$$ y = \frac{1}{2}x + 3 $$
+You can find the inverse by switching the $x$ and $y$ values and solving for $y$:
+$$ x = \frac{1}{2}y + 3 $$
+The range of the inverse is the same as the domain of the original. 
+You can verify by taking $f \circ g$, and simplifying. 
+

education/math/Standard forms of circles.md

@@ -3,7 +3,7 @@
 For $(f\circ g)(x)$ for two sets, you look for $x$ from $f$ and an equivalent $y$ value from $g$, and leftover coordinates are the answer. The order of $f$ and $g$ does matter. 
 # Formulae
 The general equation for a circle:
-$$ (x - h)^2 + (y - k)^2 =r $$
+$$ (x - h)^2 + (y - k)^2 =r^2 $$
 Distance formula:
 $$ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $$
 Midpoint foruma:

education/statistics/Correlation and Regression.md

@@ -29,9 +29,9 @@ If $x$ is above average, we expect the $y$ to be above average if there's a stro
 ## Calculating $r$ by hand
 Put the $x$ values into $L1$, put the $y$ values into $L2$. 
 
-1. Convert the $x$ each x value in the list to standard units($z$). Convert each $y$ value to standard units.
+1. Convert the $x$ each x value in the list to standard units($z$). Convert each $y$ value to standard units. This will create two new tables containing $z_x$ and $z_y$.
 $$ z = \frac{x-\bar{x}}{\sigma_x} $$
-2. Multiply the standard units for each ($x$, $y$) pair in the sets, giving you a third list, named $p$ in this example.
+2. Multiply the standard units for each ($z_x$, $z_y$) pair in the sets, giving you a fifth list, named $p$ in this example.
 $$ x * y = p$$
 3. Find the average of the values from step 3, this is $r$.
 $$ \bar{x}(p) $$
@@ -82,10 +82,29 @@ Given a scatter diagram where the average of each set lies on the point $(75, 70
 
 ### The Regression Line/Least Squared Regression Line (LSRL)
 - This line has a more moderate slope than the SD line. it does not go through the peaks of the "football"
+- Predictions can only be made if the data displays a linear association (is a football shape).
 - The regression line is *used to predict* the y variable when the x variable is given
-- The regression line also goes through the point of averages
+- In regression, the $x$ variable is the known variable, and $y$ is the value being solved for.
+- The regression line goes through the point of averages, and can be positive or negative
 $$ slope = r(\frac{\sigma_y}{\sigma_x}) $$
 - You can find the regression line by multiplying $\sigma_y$ by $r$, for the rise, then using $\sigma_x$ for the run from the point of averages.
 
-$$ z_x = \frac{x-\bar{x}}{\sigma_x} * r * \sigma_y + \bar{y} $$
+The below formula can be used to predict a y value given a 5 number summary of a set. 
-This formula finds the $z$ score for $x$, transforms by $r$, and uses the equation $x = z * \sigma + \bar{x}$ to predict a value for one axis given another axis.
+$$ \hat{y} = \frac{x-\bar{x}}{\sigma_x} * r * \sigma_y + \bar{y} $$
+1. Find $z_x$
+2. Multiply $z_x$ by $r$
+3. Multiply that by $\sigma_y$
+4. Add the average of $y$
+
+- For a positive association, for every $\sigma_x$ above average we are in $x$, the line predicts $y$ to be $\sigma_y$ standard deviations above y.x
+- There are two separate regression lines, one for predicting $y$ from $x$, and one for predicting $x$ from $y$
+- Do not extrapolate outside of the graph
+### The Regression Effect
+- In a test-retest situation, people with low scores tend to improve, and people with high scores tend to do worse. This means that individuals score closer to the average as they retest. 
+- The regression *fallacy* is contributing this to something other than chance error.
+
+---
+# Terminology
+| Term | Definition |
+| -- | -- | 
+| $\hat{y}$ | The predicted value |