# Probability distributions

# take normal as first example
# there are 4 functions
# rnorm - draw a random normal number
# pnorm - distribution function F(x)
# qnorm - quantile function F^-1(p)
# dnorm - density function f(x)
# see ?Normal


# let X be normal(0, 1)

# what's the probability X < 3
pnorm(3)
# what's the probability X > 3
1 - pnorm(3)
# or
pnorm(3, lower.tail = FALSE)


# what number is the 0.9 quantile (i.e. what z has P(X < z) = 0.9)
qnorm(0.9)

# what's a normal density look like
x <- seq(-4, 4, 0.01)
plot(x,  dnorm(x),  type = 'l')

# I need 10 random standard normals (i.e x_1, ..., x_10 iid N(0, 1))
rnorm(10)

# All take mean and sd arguments
pnorm(3, mean = 2, sd = 1)


# All other distributions follow the same format
# for example: Poisson
# rpois
# ppois
# qpois
# dpois
# take lambda argument (called theta in the book)

# other dists: try
?Chisquare
?Binomial
?Uniform
?GammaDist

# Q: What's the probability of a Normal random variable with mean 5 and variance
# 4 being greater than 2?

# Q: What's the probability of a Poisson random variable with mean 10 being 
# between 7 and 11?

# Q: Plot the density of a Gamma with scale 4 and shape 1.  Overlay a histogram of 50 random draws from the same distribution.