D05

library(tidyverse)

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library(moderndive)
df = data.frame(a = c(0,2,4,6,7,8), b = c(10,3,8,9,4,1)) %>% as_tibble()
df

1.

a. var(a) and var(b)

df %>% 
  summarise(E_a = mean(a), E_a2 = mean(a^2), E_b = mean(b), E_b2 = mean(b^2)) %>%
  summarise(var_a = E_a2 - E_a^2, var_b = E_b2 - E_b^2) -> dfvar
dfvar

b. cov(a,b)

df %>%
  summarise(E_a = mean(a), E_b = mean(b), E_ab = mean(a*b)) %>%
  summarise(cov_ab = E_ab - (E_a*E_b)) -> dfcov
dfcov

c. cor(a, b)

dfvar %>%
  summarise(std_a = sqrt(var_a), std_b = sqrt(var_b)) %>%
  bind_cols(dfcov) %>%
  summarise(cor_ab = cov_ab / (std_a*std_b))

2.

a.

cor(a,b) and cor(b,a) have the same value but opposite signs 2b.

b.

var(x) = E[x*x] - E[x]*E[x] and cov(x,y) = E[x*y] - E[x]*E[y] so they are related because you are replacing half of the references to x with y

3.

a.

cov(m, 3n) = 3*cov(m,n) = 3*16 = 48

b.

cov(m+n, m) = cov(m,n) + cov(m,m) = 16 + 25 = 41

c. 

var(m+n) = cov(m+n, m+n) = cov(m, m+n) + cov(n, m+n) = cov(m, m) + cov(m, n) + cov(n, m) + cov(n, n) = 25 + 16 + 16 + 9 = 67

d. 

cor(m,n) = cov(m,n)/sqrt(cov(m,m))*sqrt(cov(n,n)) = 16/5*6 = 2.666...

4.

tibble(x = runif(30, min=0, max=10)) %>%
  mutate(y = x^2 - 3*x + 5) -> df
df %>% head()

a.

ggplot(df, aes(x,y)) +
  geom_point() +
  geom_smooth(method = lm, se = FALSE)

## `geom_smooth()` using formula 'y ~ x'

b.

get_regression_table(lm(y~x, df))

y = 6.956x - 10.165

c. 

df %>% mutate(y_hat = 6.956*x - 10.165) -> df
df %>% head()

ggplot(df, aes(x,y)) +
  geom_point() +
  geom_smooth(method = lm, se = FALSE) +
  geom_segment(aes(xend = x, yend = y_hat), color='red', size=1.5)

## `geom_smooth()` using formula 'y ~ x'

d. 

df %>% 
  add_row(x = -10, y = (-10)^2 - 3*(-10) + 5, y_hat = 6.956*(-10) - 10.165) %>%  
  add_row(x = 20, y = 20^2 - 3*20 + 5, y_hat = 6.956*20 - 10.165) %>%
  ggplot(aes(x,y)) +
  geom_point() +
  geom_smooth(method = lm, se = FALSE) +
  geom_segment(aes(xend = x, yend = y_hat), color='red', size=1)

## `geom_smooth()` using formula 'y ~ x'