605: hw7

1.

let X1, X2, …, Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y.

each of which is uniformly distributed, which means that the probability of each X1, X2,….,Xn is a constant.

they are distributed on the integers from 1 to k, which means X1 could be {0.1,0.1,0.1,0.1,0.1….}, X2 could be {0.2,0.2,0.2,….}, … and Xn could be {1/k,1/k,1/k,1/k,…}

since the all these random variables are uniformly distributed, there will be n ways to pick the minimum value from individuals.

suppose that j is between 1 and k, the density distribution for j will be \[((k-j+1)^n - (k-j)^n)/k^n\]

2.

your organization owns a copier(future lawyers, etc.) or MRI(future doctors). This machine has a manufacturer’s expected lifetime of 10 years. This means that we expected one failure every ten years.(include the probability statements and R code for each part)

# expected one failure every ten years
# the probability of failure is 
p = 1/10

a). what is the probability that the machine will fail after 8 years?. provide also the expected value and standard deviation. Model as a geometric.(Hint: the probability is equivalent to not failing during the first 8 years..)

# geometric model: P(X > 8) = 1 - P(X <= 8)
dgeom(8, p)

## [1] 0.04304672

# expected value: 1/p
ex_geom = 1/p
ex_geom

## [1] 10

# standard deviation
sd_geom = sqrt((1-p)/p^2)
sd_geom

## [1] 9.486833

b). what is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.

# exponential mode
dexp(8,p)

## [1] 0.0449329

# expected value: 1/lambda
ex_exp = 1/p
ex_exp

## [1] 10

# standard deviation
sd_exp = sqrt(1/(p^2))
sd_exp

## [1] 10

c). what is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an binomial.(Hint: 0 success in 8 years)

# binomial model
dbinom(0,8,p)

## [1] 0.4304672

# expected value: np
ex_binom = 8 * p
ex_binom

## [1] 0.8

# standard deviation: sqrt(npq)
sd_binom = sqrt(8 * p * (1-p))
sd_binom

## [1] 0.8485281

d). what is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a Poisson.

# poisson model
lamda = 8*p
dpois(8,lamda)

## [1] 1.869665e-06

# expected value: lambda
lamda

## [1] 0.8

# standard deviation
sqrt(lamda)

## [1] 0.8944272