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probsim_tv.Rmd
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---
title: "Regression and Other Stories: Simulation"
author: "Andrew Gelman, Jennifer Hill, Aki Vehtari"
date: "`r Sys.Date()`"
output:
github_document:
toc: true
---
Tidyverse version by Bill Behrman.
Simulation of probability models. See Chapter 5 in
Regression and Other Stories.
-------------
```{r, message=FALSE}
# Packages
library(tidyverse)
# Parameters
# Common code
file_common <- here::here("_common.R")
#===============================================================================
# Run common code
source(file_common)
```
# 5 Simulation
## 5.1 Simulation of dicrete probability models
### How many girls in 400 births?
```{r}
n_births <- 400
n_sims <- 1e4
prob_girl <- 0.488
```
One observation of `r n_births` single births.
```{r}
set.seed(151)
rbinom(n = 1, size = n_births, prob = prob_girl)
```
`r format(n_sims, big.mark = ",")` observations of `r n_births` single births.
```{r}
set.seed(151)
sim_girl_1 <-
tibble(n_girls = rbinom(n = n_sims, size = n_births, prob = prob_girl))
```
Number of girls born in `r format(n_sims, big.mark = ",")` observations of `r n_births` single births.
```{r}
sim_girl_1 %>%
ggplot(aes(n_girls)) +
geom_histogram(binwidth = 2) +
labs(
title =
str_glue(
"Number of girls born in {format(n_sims, big.mark = ',')} observations of {n_births} single births"
),
x = "Number of girls born",
y = "Count"
)
```
`r format(n_sims, big.mark = ",")` observations of `r n_births` single and twin births.
```{r}
set.seed(151)
prob_fraternal <- 1 / 125
prob_identical <- 1 / 300
prob_single <- 1 - prob_fraternal - prob_identical
prob_single_girl <- 0.488
prob_fraternal_girl <- 0.495
prob_identical_girl <- 0.495
n_girls <- function(n_single, n_fraternal, n_identical) {
n_single_girls <- rbinom(n = 1, size = n_single, prob = prob_single_girl)
v <-
rmultinom(
n = 1,
size = n_fraternal,
prob =
c(
(1 - prob_fraternal_girl)^2,
2 * (1 - prob_fraternal_girl) * prob_fraternal_girl,
prob_fraternal_girl^2
)
) %>%
t() %>%
as_tibble(.name_repair = ~ c("n_girls_0", "n_girls_1", "n_girls_2"))
n_fraternal_girls <- v$n_girls_1 + 2L * v$n_girls_2
n_identical_girls <-
2L * rbinom(n = 1, size = n_identical, prob = prob_identical_girl)
n_single_girls + n_fraternal_girls + n_identical_girls
}
sim_girls_2 <-
tibble(
n_girls =
rmultinom(
n = n_sims,
size = n_births,
prob = c(prob_single, prob_fraternal, prob_identical)
) %>%
t() %>%
as_tibble(
.name_repair = ~ c("n_single", "n_fraternal", "n_identical")
) %>%
pmap_int(n_girls)
)
```
Number of girls born in `r format(n_sims, big.mark = ",")` observations of `r n_births` single and twin births.
```{r}
sim_girls_2 %>%
ggplot(aes(n_girls)) +
geom_histogram(binwidth = 2) +
labs(
title =
str_glue(
"Number of girls born in {format(n_sims, big.mark = ',')} observations of {n_births} single and twin births"
),
x = "Number of girls born",
y = "Count"
)
```
## 5.2 Simulation of continuous and mixed discrete / continuous models
```{r}
mean <- 3
sd <- 0.5
```
Normal distribution with mean `r mean` and standard deviation `r sd`.
```{r}
set.seed(660)
v <- tibble(x = rnorm(n_sims, mean = mean, sd = sd))
v %>%
ggplot(aes(x)) +
geom_histogram(binwidth = 0.1) +
labs(
title =
str_glue(
"Normal distribution with mean {mean} and standard deviation {sd}"
)
)
```
Corresponding log-normal distribution.
```{r}
v %>%
ggplot(aes(exp(x))) +
geom_histogram(binwidth = 2) +
labs(title = "Corresponding log-normal distribution")
```
```{r}
size <- 20
prob <- 0.6
```
Binomial distribution with `r size` trials and probability `r prob`.
```{r}
set.seed(136)
v <- tibble(x = rbinom(n_sims, size = size, prob = prob))
v %>%
ggplot(aes(x)) +
geom_bar() +
scale_x_continuous(breaks = seq(0, size), minor_breaks = NULL) +
coord_cartesian(xlim = c(0, size)) +
labs(
title =
str_glue(
"Binomial distribution with {size} trials and probability {prob}"
)
)
```
```{r}
lambda <- 5
```
Poisson distribution with mean `r lambda`.
```{r}
set.seed(185)
v <- tibble(x = rpois(n_sims, lambda = lambda))
v %>%
ggplot(aes(x)) +
geom_bar() +
scale_x_continuous(breaks = seq(0, max(v$x)), minor_breaks = NULL) +
labs(title = str_glue("Poisson distribution with mean {lambda}"))
```
Simulation parameters on probabilities and heights of adults.
```{r}
n_sims <- 1000
prob_men <- 0.48
height_men_mean <- 69.1
height_men_sd <- 2.9
height_women_mean <- 64.5
height_women_sd <- 2.7
```
Mean height of 10 randomly chosen adults.
```{r}
height_mean <- function(n = 10) {
if_else(
rbinom(n, size = 1, prob = prob_men) == 1,
rnorm(n, mean = height_men_mean, sd = height_men_sd),
rnorm(n, mean = height_women_mean, sd = height_women_sd)
) %>%
mean()
}
```
One observation.
```{r}
set.seed(815)
height_mean()
```
`r format(n_sims, big.mark = ",")` observations.
```{r}
set.seed(815)
sim_height_mean <- tibble(height_mean = map_dbl(seq_len(n_sims), height_mean))
```
Distribution of the mean height of 10 adults.
```{r}
sim_height_mean %>%
ggplot(aes(height_mean)) +
geom_histogram(binwidth = 0.1) +
labs(title = "Distribution of the mean height of 10 adults")
```
Maximum height of 10 randomly chosen adults.
```{r}
height_max <- function(n = 10) {
if_else(
rbinom(n, size = 1, prob = prob_men) == 1,
rnorm(n, mean = height_men_mean, sd = height_men_sd),
rnorm(n, mean = height_women_mean, sd = height_women_sd)
) %>%
max()
}
```
One observation.
```{r}
set.seed(815)
height_max()
```
`r format(n_sims, big.mark = ",")` observations.
```{r}
set.seed(815)
sim_height_max <- tibble(height_max = map_dbl(seq_len(n_sims), height_max))
```
Distribution of the maximum height of 10 adults.
```{r}
sim_height_max %>%
ggplot(aes(height_max)) +
geom_histogram(binwidth = 0.5) +
scale_x_continuous(breaks = scales::breaks_width(1)) +
labs(title = "Distribution of the maximum height of 10 adults")
```