---
title: 'Beamer presentation using R Markdown'
author: "Vincenzo Gioia"
date: "08/10/2024"
lang: "it"
output:
  beamer_presentation:
    theme: Madrid
    colortheme: default
    slide_level: 2
    includes:
      in_header: setBeamer.tex
  ioslides_presentation: default
---

```{r setup, include=FALSE}
#knitr::opts_chunk$set(echo = FALSE)
def.source.hook  <- knitr::knit_hooks$get("source")
```

```{r,echo=FALSE,message=FALSE,warning=FALSE}
library(knitr)
library(pander)
# Set so that long lines in R will be wrapped:
opts_chunk$set(tidy.opts=list(width.cutoff=80),tidy=TRUE)
```

```{r set-options, echo=FALSE, cache=FALSE}
options(width = 60)
```

\tableofcontents

# Central Limit Theorem (CLT)
Let $X_1, X_2, \ldots, X_n$,  be a sequence of independent and identically distributed (iid) random variables (rv) from a distribution with mean $\mu$ and finite variance $\sigma^2$.

For large n,
$$   \bar X_n = \frac{1}{n} \sum_{i=1}^{n}X_i \, \dot \sim \, \mathcal{N}(\mu, \sigma^2/n)$$
where $\dot \sim$ indicates convergence in distribution.
Equivalently $$S_n = \sum_{i=1}^{n}X_i \, \dot \sim \, \mathcal{N}(n\mu, n\sigma^2)$$
The CLT supports the normal approximation to the distribution of a rv that can be viewed as the sum of other rv.

## Approximation with CLT: application

The approximation above is useful in statistics for computing some quantities. For instance, let $X$ and $Y$ be two independent Binomial rv, such that $X \sim {\rm Bin}(n,p)$ and $Y \sim {\rm Bin}(m, q)$.

If we are interested in computing the probability $P(X>Y)$ the Normal approximation is the simplest way to compute it. We can approximate: $$X \, \approx \, {\cal N}(np, np(1-p)), \quad \quad \quad \ Y \, \approx \, {\cal N}(mq, mq(1-q)).$$

Then, by using a well known probability result, the difference $W=X-Y$ of two independent normal distributions with means $\mu_X, \mu_Y$ and variances $\sigma^2_X, \sigma^2_Y$, respectively, is **still a normal distribution** with mean $\mu_W=\mu_X-\mu_Y$ and variance $\sigma^2_W=\sigma^2_X+\sigma^2_Y$.

In such a case, $$\mu_W=\mu_X-\mu_Y= np-mq, \quad \quad \quad \sigma^2_W=\sigma^2_X+\sigma^2_Y=np(1-p)+mq(1-q).$$


# CLT application to waterpolo goals

Tomorrow two professional Italian waterpolo teams, Posillipo and Pro Recco, compete against each other. Let $X$ and $Y$ be the random *goals scored* by Posillipo and Pro Recco, respectively.

We assume that $X, Y$ follow two independent Binomial distributions. Thus, $X$ and $Y$ represent the number of shots converted in goal on the total number of shots $n, m$ made by Posillipo and Pro Recco, with probabilities $p$ and $q$, respectively.

Before the match, the number of shots is *unknown*. In what follows, we adopt a simplification, and we treat the quantities $p,q,m,n$ as *known*, for instance fixing them upon historical experience: $p=0.5, q=0.7, n=20, m=20$.

We want to investigate the Posillipo probability of winning the next match against Pro Recco, that is $$P(X>Y)=P(X-Y>0)=?$$

## CLT application to waterpolo goals

So, let $W$ be the r.v., such that $W=X-Y$. We could compute the law of this rv but using the Normal approximation, $W \approx \mathcal{N}(\mu_W = \mu_X-\mu_Y, \sigma^2_W = \sigma^2_X + \sigma^2_Y)$, we can esily compute such a probability of interest. 

```{r binom_app, echo=TRUE, eval=TRUE, results='hold', fig.keep='high', fig.height=8, fig.width=5}
p <- 0.5
q <- 0.7 
n <- m <- 20
mW <-  p * n - q * m
sdW <- sqrt( n * p * (1 - p) + m * q * (1 - q) )
# Probability that  W = X-Y > 0 (Posillipo win the match)
PWin_P <- pnorm(0, mean = mW, sd = sdW, lower.tail = FALSE)
PWin_P

```

## CLT application to waterpolo goals
Here, we show the probability mass functions of $X$ and $Y$ and the probability density function of $W=X-Y$. 

\footnotesize
```{r binom_app_plot, echo=TRUE, eval=FALSE}
# pdf of W
curve(dnorm(x, mW, sdW),
      xlim = c(-12, 20), ylim = c(0, 0.3), 
      xlab = "", ylab = "", cex.lab = 1.25)

# pmf of X and Y 
points(0 : 20, dbinom(0 : n, n, p), pch = 21, bg = 1, col = "blue")
points(0 : 20, dbinom(0 : m, m, q), pch = 21, bg = 2)

segments(0, -0.01,              #(x_0, y_0)
         0, dnorm(0, mW, sdW),  #(x_1, y_1)
         lwd = 2)

text(14.25, 0.22, "Y", cex = 2, col = 2)
text(10, 0.2, "X", cex = 2, col = "blue")
text(-4, 0.15, "X - Y", cex = 2, col = 1)
```

## CLT application to waterpolo goals

```{r binom_app_plot2, echo=FALSE, eval =TRUE}
# pdf of W
curve(dnorm(x, mW, sdW),
      xlim = c(-12, 20), ylim = c(0, 0.3), 
      xlab = "t", ylab = "", cex.lab = 1.25)

# pmf of X and Y 
points(0 : 20, dbinom(0 : n, n, p), pch = 21, bg = 1, col = "blue")
points(0 : 20, dbinom(0 : m, m, q), pch = 21, bg = 2)

segments(0, -0.01,              #(x_0, y_0)
         0, dnorm(0, mW, sdW),  #(x_1, y_1)
         lwd = 2)

text(14.25, 0.22, "Y", cex = 2, col = 2)
text(10, 0.2, "X", cex = 2, col = "blue")
text(-4, 0.15, "X - Y", cex = 2, col = 1)
```