Problem 10.1.3 Posterior Sampling.Rmd

---
title: "Problem 10From: Ben Lambert. “A Student’s Guide to Bayesian Statistics”."
output: html_notebook
---

```{r}
fPosteriorPredictive <- function(aNumSamples){
  lX <- vector(length=aNumSamples)
  for(i in 1:aNumSamples){
    theta <- rgamma(1, 3+35, 0.5+5)
    X <- rpois(1,  theta)
    lX[i] <- X
}
return(lX) }
```

```{r}
X <- fPosteriorPredictive(10000)
hist(X, breaks=seq(0, 100, 1), xlim = c(0, 20),
     xlab="10,000 posterior predictive samples")
```
```{r}
mean(X<=3)
```
```{r}
mean(X>=11)
```
```{r}
mean(X>=20)
```
“Problem 10.1.7 By using your previously determined posterior as a prior, update your posterior to reflect the new datum. Graph the PDF for this new distribution.”
```{r}
curve(dgamma(x, 3+35+20, 0.5+5+1), 0, 20, xlab = "lambda", ylab = "pdf")
```
```{r}
fPosteriorPredictive <- function(aNumSamples){
  lX <- vector(length=aNumSamples)
  for(i in 1:aNumSamples){
    theta <- rgamma(1, 3+35+20, 0.5+5+1)
    X <- rpois(1,  theta)
    lX[i] <- X
}
return(lX) }

X <- fPosteriorPredictive(10000)
hist(X, breaks=seq(0, 100, 1), xlim = c(0, 20),
     xlab="10,000 posterior predictive samples")

```
```{r}
mean(X>=20)
```