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### EXERCISE 3.1 NORMAL DISTRIBUTION
################
### A)
### Let the weight distribution of a production series
### of a tinned product be normally distributed with
### expectation mu and standard deviation sigma.
### Plot the distribution function for different mu and 
### variance sigma^2. Use additional graphic parameters
### you know from Exercises_part2, e.g. xlab, cex.lab, etc.
mu = 800
sigma2 = 100
x = seq(800-200, 800+200, 1)
y = dnorm(x, mean=mu, sd=sqrt(sigma2))
plot(x, y, type="l")

### B)
### Given mu=800g and sigma2=100, determine the probability
### that a weight of an individual piece of the production
### is less or equal tau=790g. You can use the cumulative distribution
### function of the normal distribution.
tau = 790
res = pnorm(tau, mean=800, sd=10)
res
x = seq(800-30, 800+30, 1)
y = pnorm(x, mean=mu, sd=10)
plot(x, y, type="l")
lines(c(tau, tau), c(0, res), type="l", lty=2)
lines(c(0, tau), c(res, res), type="l", lty=2)
### What is the probability that a selected piece has a weight
### of more than tau?

### C)
### Draw a random sample from your production using different
### sample sizes n. Determine mean, sd and quartiles (25%, 50%, 75%-quantiles)
### of your sample.
### Compare estimates from the sample and true values of the distribution.
mu = 800
sigma = 10
n = 10
S = rnorm(n=n, mean=mu, sd=sigma)
S
### Estimates from the sample:
summary(S)
var(S)
sd(S)
### True quantiles
qnorm(p=c(.25, .5, .75), mean=mu, sd=sigma)


################
### EXERCISE 3.2 BINOMIAL DISTRIBUTION
################
### A)
### Consider you sample n pieces from your production.
### Further, assume the production process is configured
### such that the probability that a particular piece is
### useable is p (useable means the piece can be sold).
### Draw the probability function for different n and p.
n = 100
p = 0.8
x = seq(0, 100, 1)
y = dbinom(x, size=n, prob=p)
plot(x, y, type="h", xlab="Number of useable pieces", ylab="Probability")

### B)
### Given n=100 and p=0.8.
### Determine the probability that exactly 90 pieces of your
### production can be used.
dbinom(90, size=100, prob=0.8)
plot(x, y, type="h", xlab="Number of useable pieces", ylab="Probability")
points(c(90, 90), c(0, dbinom(90, size=100, prob=0.8)), col=2, type="l", lwd=2)
### Determine the probability that less than 90 pieces are usable.
### Use the cumulative distribution function.
pbinom(89, size=100, prob=0.8)
res = pbinom(89, size=100, prob=0.8)
x = seq(0, 100, 1)
y = pbinom(x, size=n, prob=p)
plot(x, y, xlab="Number of useable pieces", ylab="Cumulative Probability")
lines(c(89, 89), c(0, res), type="l", lty=2)
lines(c(0, 89), c(res, res), type="l", lty=2)
### Attention: the cumulative distribution function
### of a discrete variable is usually a step function.

### C)
### Simulate the following scenario:
### Consider you draw each day of the year a random sample of n=100 pieces
### of your production as a quality control, and study the number of useable pieces.
### Your production is anticipated to have a rate of 97% of useable pieces.
### Determine mean and quartiles (25%, 50%, 75%-quantiles) of your sample.
### Compare estimates from the sample and true values of the distribution.
n = 100
p = 0.97
S = rbinom(365, size=n, prob=p)
S
### Estimates from the samples.
### Remark: estimates need to be devided by n to obtain probabilities
### in the range of [0, 1].
summary(S)/n
### True quantiles.
qbinom(p=c(.25, .5, .75), size=n, prob=p)/n

### Since quality control is expensive, try out whether smaller n yield sufficient
### estimates.


################
### EXERCISE 3.3
################
### Draw density function and cumulative distribution functions
### for the Poisson distribution and the Exponential distribution.
### Follow Examples 9 and  10 from the lecture.
### You can use the function dpois, dexp, etc.
### Use the functions rpoins and rexp to draw random data from the
### both distributions.