# Binomial Distribution

# In an inspection of automobiles in Los Angeles, 60%
# had emissions that did not meet EPA regulations.
# For a random sample of 10 automobiles, 
# let X=# vehicles failing inspection
# find the following probabilities

# a) all 10 vehicles failed the inspection
# b) Exactly 6 of the 10 failed the inspection
# c) Six or more failed the inspection
# d) All 10 passed the inspection

# X~bin(n,p)=X~bin(10,0.6)

# P(X=x)=(nCr)(p^x)(q^(n-x)) where q=1-p
# dbinom(x,n,p)

# a) P(X=10)
dbinom(10,10,.6)

# b) P(X=6)
dbinom(6,10,.6)

#c) P(X>=6)=P(6)+P(7)+P(8)+P(9)+P(10)
sum(dbinom(6:10,10,.6))

#d) P(X=10 "failures") where p= 1-.6 =.4
dbinom(10,10,.4)
# OR P(X=0 "successes") where p=.6
dbinom(0,10,.6)

# hypergeometric function dhyper(x,M,N-M,n)

# Five individuals from an animal population thought to be near
# extinction in a certain region have been caught, tagged, and released
# to mix into the population. After they have had an opportunity to
# mix, a random sample of 10 of these animals is selected. Let X =
# the number of tagged animals in the second sample. If there are
# actually 25 animals of this type in the region, find the following:

M=5
N=25
n=10


# Probability that exactly 2 are tagged? (P(X = 2))
# Probability that exactly 4 are tagged? (P(X = 4))
# Probability that at most 2 are tagged? (P(X <= 2))
# Find EX, VX, SDX

# P(X=2)

dhyper(2,M,N-M,n) # or dhyper(2,5,25-5,10)

# P(X=4)

dhyper(4,M,N-M,n) # or dhyper(4,5,25-5,10)

# P(X <= 2)

sum(dhyper(0:2,M,N-M,n)) # or sum(dhyper(0:2,5,25-5,10))

# EX=n(M/N)  VX=n(M/N)(1-M/N)((N-n)/(N-1))=EX*(1-M/N)((N-n)/(N-1))
# bound is the bound on the error of estimation, aka Margin of error

EX=n*(M/N); EX
VX=EX*(1-M/N)*((N-n)/(N-1)); VX
SDX=sqrt(VX); SDX
rbind(EX,VX,SDX)

# Poisson distribution


# Cars arrive at a toll booth on average at a rate of 6 cars per 10 seconds
# during rush hours.  Let X be the number of cars arriving
# during any 10-second period during rush hours.
# Find the following probabilities:

# a) No cars arrive
# b) More than one car arrives
# c) At least 2 cars arrive

# X~poi(lambda)=X~poi(mu)=X~poi(6)
# use dpois(x,lambda)

# a) P(X=0) no cars arrive in next 10 seconds
dpois(0,6)

# b) P(X>1)=1-P(X<=1)
1-sum(dpois(0:1,6))

# c)P(X>=2)=1-P(X<2)=1-P(X<=1)
1-sum(dpois(0:1,6))


# Geometric distribution

# An Olympic archer is able to hit the bull's-eye 80% of the time
# Assume each shot is independent of one another
# Find:

# a) probability the first bull's-eye is on the 2nd shot
# b) probability the first bull's-eye is on the 3rd or 4th shot
# c) would you be surprised if the 1st one came on the 6th shot? 
# calculate probability the first one is on the 6th shot
# d) mean, variance, standard deviation

# a)

dgeom(2,.8)

# b) 

sum(dgeom(3:4,.8))

# c) 

dgeom(6,.8) # should be small so very surprised!

# d) 
EX=1/.8; EX
VX=.2/(.8^2); VX
SDX=sqrt(VX); SDX