A discrete random variable $X$ has possible values 0, 1, 2, 3, 4, 5, 6, 7, 8 with the following partial probability distribution. The missing probabilities $\prob(X=7)$ and $\prob(X=8)$ are equal to each other.

```{r, echo = FALSE} set.seed(20221021) x = 0:8 p = rnorm(7,5,2.1) p = sort(round(p / sum(p) * 0.84, 2)) p1 = tibble(x = x, p = c(p, rep(NA,2)))

p1_tab = p1 %>% mutate(x = str_c(x)) %>% pivot_wider(names_from = x, values_from = p) %>% mutate(x = "P(X=x)", .before = `0`)

p1_tab %>% kable() %>% kable_styling(position = "center", full_width = FALSE)

Calculate the mean, the variance, and the standard deviation of this distribution.