riddler-spam-comments.Rmd

---
title: "Riddler: Spam Comments"
output: html_document
---

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

```{r}
library(tidyverse)

sim <- replicate(1e6, which(cumsum(rexp(250, 1:250)) > 3)[1] - 1)

mean(sim, na.rm = TRUE)
```

```{r}
comment_counts <- tibble(n_comments = sim) %>%
  count(n_comments) %>%
  mutate(density = n / sum(n))

library(broom)

augmented_exp <- nls(density ~ lambda * exp(-lambda * n_comments),
                     data = comment_counts,
                     start = list(lambda = 1)) %>%
  augment(comment_counts)

augmented_geometric <- nls(density ~ (1 - p) ^ n_comments * p,
                           data = comment_counts,
                           start = list(p = .05)) %>%
  augment(comment_counts)

augmented_geometric %>%
  ggplot(aes(n_comments, density)) +
  geom_line() +
  geom_line(aes(y = .fitted), color = "red")
```

The number of comments after 3 days is described by a geometric distribution.

What determines the parameter $p$ (and therefore the expected value) of the geometric distribution?


```{r}
tidy_sim <- crossing(trial = 1:5e4,
         step = 1:250) %>%
  mutate(waiting = rexp(n(), step)) %>%
  group_by(trial) %>%
  mutate(cumulative = cumsum(waiting)) %>%
  ungroup()

ncomments <- tidy_sim %>%
  mutate(within_3 = cumulative < 3) %>%
  group_by(trial) %>%
  summarize(n_comments = sum(within_3))
```

```{r}
comments_by_threshold <- tidy_sim %>%
  crossing(threshold = seq(.25, 3, .25)) %>%
  mutate(within = cumulative < threshold) %>%
  group_by(threshold, trial) %>%
  summarize(n_comments = sum(within))

comments_by_threshold %>%
  summarize(expected_value = mean(n_comments)) %>%
  ggplot(aes(threshold, expected_value)) +
  geom_line() +
  geom_line(aes(y = exp(threshold) - 1), color = "red") +
  labs(x = "# of days",
       y = "Expected number of comments",
       title = "How many comments cumulatively over time?",
       subtitle = "Red line shows exp(x) - 1")

comment_counts %>%
  filter(!is.na(n_comments)) %>%
  ggplot(aes(n_comments, density)) +
  geom_line() +
  geom_line(aes(y = (1 - 1 / exp(3)) ^ n_comments / exp(3)), color = "red")
```

The number of comments after n days is described by a geometric distribution with expected value $e^{n}-1$ (that is, with success probability $exp(-n)$).