######################################################################
# random sample from the normal distribution, estimate the parameters
# simulated sample

mu = 43; sigma = 5.97   # parameters
n = 300

x = rnorm(n, mu, sigma)

hist(x,freq=FALSE)

# estimate the parameters with x.bar and sd

mu.hat = mean(x); mu.hat
sigma.hat = sd(x); sigma.hat

# plot the fitted model on the histogram

curve(dnorm(x,mu.hat,sigma.hat),add=T)

######################################################################
# random sample from the exponential distribution, estimate the parameters
# simulated sample

lambda = 11.34   # parameters
n = 30

x = rexp(n, rate=lambda)

hist(x,freq=FALSE)

# estimate the parameters with x.bar and sd

lambda.hat = 1/mean(x); lambda.hat

# plot the fitted model on the histogram

curve(dexp(x,lambda.hat),add=T)

######################################################################
# now suppose we want to fit the gamma distribution to a random sample
# how can we do this?  can we use x.bar and the sd?

# The big question is, how do we estimate parameters in models in general?

######################################################################