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Author: cuadrosangeles

labs/Lab_4.Rmd

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+---
+title: "Lab 4"
+date: '10/21/2022'
+output:
+  pdf_document: default
+  html_document: default
+---
+
+
+```{r setup, include=FALSE}
+knitr::opts_chunk$set(echo = TRUE)
+```
+
+# The dangers of hard coding
+
+One of the most common problems from the last problem set was in question 1e, which asked you to write a function called mymean(). Many people had the correct solution, or something which worked similarly:
+
+```{r}
+mymean <- function(vec){
+  sum(vec)/length(vec)
+}
+```
+
+| $x$  | $P(X = x)$ |
+|------|------------|
+|  0   |     1/8    |
+|   1  |    3/8     |
+|   2  |     3/8    |
+|   3  |    1/8     |
+
+
+We can code this distribution like this:
+
+```{r}
+X <- c(0, 1, 2, 3)
+probs <- c(1/8, 3/8, 3/8, 1/8)
+```
+
+Then we can sample from it like this:
+
+```{r}
+set.seed(2418)
+
+sample_1 <- sample(x = X,
+                   size = 1,
+                   prob = probs)
+sample_1
+```
+
+```{r}
+sample_10000 <- sample(x = X,
+                       size = 10000,
+                       prob = probs,
+                       replace = TRUE)
+
+sample_10000
+```
+
+Now, back to our function. Let’s check to make sure it works:
+
+```{r}
+mean(sample_10000)
+```
+```{r}
+mymean(sample_10000)
+```
+
+Many assignments that something was “hard-coded,” and that even though the function may have produced the correct answer in this case, it would not work as intended in general.
+
+First, consider this version of mymean:
+
+```{r}
+n <- 10000
+
+mymean_2 <- function(vec){
+  sum(vec)/length(vec)
+}
+
+mymean_2(sample_10000)
+```
+
+It looks like it works, but notice that the denominator is n instead of length(vec). In this case, the denominator has been “hard-coded” into the function. It does not change when we pass in a different vector. The downside of the hard-coding is that the function only works correctly if you pass in a vector with 10000 entries.
+
+```{r}
+sample_5000 <- sample(x = X,
+                      size = 5000,
+                      prob = probs,
+                      replace = TRUE)
+
+mymean_2(sample_5000)
+
+```
+This value is roughly half of what we would expect. Why?
+
+To be clear, there isn’t anything wrong with mymean_2. It correctly calculates the mean under certain conditions (specifically, when you have exactly 10000 data points). But, it’s usefulness is farily limited. This
+is almost always true when you hard-code values, parameters, etc. in your functions, so it is a practice to avoid if possible (and it usually is).
+
+
+# Conditional Probability of two Random Variables
+
+When writing the PMF of X conditional on Y, X is the random variable we defined above and Y is a random variable which takes on the value 1 if all three flips are heads and 0 otherwise. 
+
+Conditional probability is a concept that will come up again and again when we talk about inference and regression.
+
+| $x$  |$y$ | $P(X = x | Y = y)$ |
+|------|----|---------------------|
+|   0  | 0  |              1/7    |
+|   1  | 0  |              3/7    |
+|   2  | 0  |              3/7    |
+|   3  | 1  |              1      |
+
+One thing that always helps me conceptualize conditional probabilities is to read | as “assuming that. . . ” So, if we want to know P(X = x|Y = y), we want the “probability that X = x assuming that Y = y.
+
+First, assume that Y = 1. There is only one possible event that maps to this value of Y : {HHH}. Therefore, the probability that X = 3 given that Y = 1 is one, and the probability that X takes any other value is 0. Now, assume that Y = 0. How many events map to this value of Y ? Seven: {T T T, T TH, T HH, HT T, HT H, T HT, HHT }.
+
+Now, calculating the conditional probabilities in the table above is as simple as counting how many of the seven map into particular values of X. There is one for which X = 0, 3 for which X = 1, and 3 for which X = 2.
+
+# Final Projects!
+
+Hopefully everyone has had a chance to locate an interesting data set you want to work with. We are nearly halfway through the quarter and I want to make sure that everyone is making progress. Hopefully this week’s material has prompted deeper thinking about what your data can tell you.
+
+Pull up your data and briefly describe it to the person sitting next to you. Where does it come from? What is the unit of observation?
+
+If you haven’t chosen a dataset, what are you considering? What topics interest you? See if anyone around you has suggestions.
+
+Have you run into any problems wrangling your data? Discuss ongoing challenges with the people around you.
+
+What are some questions your data can help you answer? Are they causal or descriptive questions?
+