Stats.Rmd

---
title: "Stats!"
author: "Gregory J. Matthews"
date: "2023-11-16"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

Topics: F-test and t-test with regards to regression Propensity score matching RMSE and predicted vs actual values.\
Design of Experiments and Split plot designs.

## Probability

-   What is it?\
-   Some rules

$P(A^c) = 1 - P(A)$

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$P(A \cap B) = P(A) P(B)$ if $A \perp B$

$P(A) = P(A|B)P(B) + P(A|B^c)P(B)$

$P(A | B) = \frac{P(A\cap B)}{P(A)}$

$P(A\cap B) = P(A|B)P(B)$

$P(B | A) = \frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)}$

## Distributions

-   pdf vs cdf
-   Normal
-   Binomial
-   Others

CDF of X: $F(x) = P(X \le x)$

PDF of X: $f(x) = P(X = x)$ ONLY IN DISCRETE CASE!

```{r}
x <- seq(-4,4,0.01)
plot(x, dnorm(x), type = "l", main = "pdf Normal")
plot(x, pnorm(x), type = "l", main = "cdf Normal")
```

```{r}
x <- c(0:10)
plot(x, dbinom(x,10,0.5),  main = "pdf binomial")
x <- seq(0,10,0.01)
plot(x, pbinom(x,10,0.5), type = "l",  main = "cdf binomial")
```

```{r}
# Let X ~ normal(0,1)
# Let Y ~ binomial(10,0.5)
# Let B ~ binomial(1,0.5)
# Let Z | B ~ normal(B, 1)
#Find probability that X > 1 give X > 0.  
#That is find P(X > 1 | X > 0). 

#Find P(Y = 5 AND -1 < X < 1)

# Find P(Z > 0)

```

## Inference

-   Point Estimates
-   Confidence intervals
-   Hypothesis testing
    -   Permutation testing
    -   t-test
    -   $\chi^2$-squared test
    -   p-values!

```{r}
#Lets do some simulations about point estimates and look at their properties 
#sample mean vs sample median

#Let's look at confidence interval coverage for a proportion. 

# Let's do hypothesis testing "by hand".  We'll do a t-test by hand.

#Let's do a permutation test

# Let's talk about p-values.  



```

```{r}
#Two simulation studies 
#1 hypothesis testing 
set.seed(2023)
mu <- 49 
sigma <- 5
n <- 20 
mu0 <- 49
nsim <- 100000
null <- rep(NA, nsim)

for (i in 1:nsim) {
  print(i)
  d <- rnorm(n, mu, sigma)
  null[i] <- (mean(d) - mu0) / (sd(d) / sqrt(n))
}
hist(null)

mean(null < 0)
mean(null < 1)
mean(null < 2)
mean(null < 3)

pt(0,19)
pt(1,19)
pt(2,19)
pt(3,19)

pnorm(0)
pnorm(1)
pnorm(2)
pnorm(3)



#





#2 prop CI





```