## Settings for RMarkdown http://yihui.name/knitr/options#chunk_options
opts_chunk$set(comment = "", warning = FALSE, message = FALSE, tidy = FALSE,
echo = TRUE, fig.width = 7, fig.height = 7)
options(width = 116, scipen = 10)
setwd("~/statistics/bio201/")
library(ggplot2)
| function type | What it does | -norm (normal distribution) | -binom (binomial distribution) |
|---|---|---|---|
| d- | probability density (mass): y value given x value | dnorm | dbinom |
| p- | probability given x value | pnorm | pbinom |
| q- | x value given probability | qnorm | qbinom |
Normal distribution
## Density at 90 in a normal distribution with a mean of 124 and sd of 20
dnorm(x = 90, mean = 124, sd = 20)
[1] 0.004702
## Density at c(10,20,30) in a normal distribution with a mean of 124 and sd of 20
dnorm(x = c(10,20,30), mean = 124, sd = 20)
[1] 0.000000001757 0.000000026805 0.000000318491
## Graphing the distribution: Do not give a value to x, then wrap with curve() function
curve(dnorm(x, mean = 124, sd = 20), xlim = c(0, 200))
abline(h = 0)
Binomial distribution
## Density (mass) at 3 in a binomial distribution with 5 trials with success probability of 0.4
dbinom(x = 3, size = 5, prob = 0.4)
[1] 0.2304
## Density (mass) at c(1,2,3) in a binomial distribution with 5 trials with success probability of 0.4
dbinom(x = c(1,2,3), size = 5, prob = 0.4)
[1] 0.2592 0.3456 0.2304
## Graphing the distribution: Use barplot() as it is a discrete distribution. Give 0:5 (= c(0,1,2,3,4,5))
barplot(dbinom(x = 0:5, size = 5, prob = 0.4), names.arg = 0:5)
Probability function (p-): Given an x value, it returns the probability (AUC) of having a value lower than x.
Quantile function (q-): Given a probability (AUC), it returns the x value at the upper boundary.
These two are doing the opposite actions.
Normal distribution
## Probability of having a value *lower* than 90 in a normal distribution with a mean of 124 and sd of 20.
pnorm(q = 90, mean = 124, sd = 20, lower.tail = TRUE)
[1] 0.04457
curve(dnorm(x, mean = 124, sd = 20), xlim = c(0, 200))
abline(h = 0)
sequence <- seq(0, 90, 0.1)
polygon(x = c(sequence,90,0),
y = c(dnorm(c(sequence),124,20),0,0),
col = "grey")
## Probability of having a value *higher* than 140 in a normal distribution with a mean of 124 and sd of 20.
1 - pnorm(q = 140, mean = 124, sd = 20, lower.tail = TRUE)
[1] 0.2119
pnorm(q = 140, mean = 124, sd = 20, lower.tail = FALSE)
[1] 0.2119
curve(dnorm(x, mean = 124, sd = 20), xlim = c(0, 200))
abline(h = 0)
sequence <- seq(140, 200, 0.1)
polygon(x = c(sequence,200,140),
y = c(dnorm(c(sequence),124,20),0,0),
col = "grey")
## X-axis value *below* which the probability (AUC) is 10%
qnorm(p = 0.10, mean = 124, sd = 20, lower.tail = TRUE)
[1] 98.37
curve(dnorm(x, mean = 124, sd = 20), xlim = c(0, 200))
abline(h = 0)
sequence <- seq(0, 98.36897, 0.1)
polygon(x = c(sequence,98.36897,0),
y = c(dnorm(c(sequence),124,20),0,0),
col = "grey")
## X-axis value *above* which the probability (AUC) is 20%
qnorm(p = 0.10, mean = 124, sd = 20, lower.tail = FALSE)
[1] 149.6
curve(dnorm(x, mean = 124, sd = 20), xlim = c(0, 200))
abline(h = 0)
sequence <- seq(149.631, 200, 0.1)
polygon(x = c(sequence,200,149.631),
y = c(dnorm(c(sequence),124,20),0,0),
col = "grey")
Binomial distribution
Definitions are confusing. It is probably safer to use summation of dbinom.
## Probability of having a value *lower* than or equal to 3 in a binomial distribution with 5 trials with success probability of 0.4. P[X <= x]. Include the "equal to" value.
pbinom(q = 3, size = 5, prob = 0.4, lower.tail = TRUE)
[1] 0.913
barplot(dbinom(x = 0:5, size = 5, prob = 0.4), names.arg = 0:5, col = rep(c("grey","white"), c(4,2)))
## Probability of having a value *higher* than 3 in a binomial distribution with 5 trials with success probability of 0.4. P[X > x]. Be careful it excludes the probability at 3.
pbinom(q = 3, size = 5, prob = 0.4, lower.tail = FALSE)
[1] 0.08704
barplot(dbinom(x = 0:5, size = 5, prob = 0.4), names.arg = 0:5, col = rep(c("white","grey"), c(4,2)))
## Summation of individual probabilities may be safer.
sum(dbinom(x = 3:5, size = 5, prob = 0.4))
[1] 0.3174
## X-axis value *below (including the value)* which the summation of probabilities is greater than or equal to p.
qbinom(p = 0.3, size = 5, prob = 0.4, lower.tail = TRUE)
[1] 1
barplot(dbinom(x = 0:5, size = 5, prob = 0.4), names.arg = 0:5, col = rep(c("grey","white"), c(2,4)))
## Summation of individual probabilities may be safer.
cumsum(dbinom(x = 0:5, size = 5, prob = 0.4))
[1] 0.07776 0.33696 0.68256 0.91296 0.98976 1.00000
## Summation in the opposite direction.
cumsum(rev(dbinom(x = 0:5, size = 5, prob = 0.4)))
[1] 0.01024 0.08704 0.31744 0.66304 0.92224 1.00000