- 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 .
expanded solution
- 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 expect one failure every ten years. (Include the probability statements and R Code for each part.).
geometric distribution video
- 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..)
p = 1/10
q = 1-p
n <- 8
pgeom(8,p,lower.tail = F)
## [1] 0.3874205
Expected value:
p^-1
## [1] 10
Standard deviation:
formulas for geometric distribution
variance = (1-p)/p^2
sd = ((1-p)/p2).05
((1-p)/p^2)^.05
## [1] 1.252311
- 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 distribution video
mean = 10
r = 1/mean
exp(-8*r)
## [1] 0.449329
Expected value:
1/r
## [1] 10
Standard deviation:
- same as the expected value
1/r
## [1] 10
- What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as a binomial. (Hint: 0 success in 8 years)
dbinom(0,n,p)
## [1] 0.4304672
Expected value:
n*p
## [1] 0.8
Standard deviation:
sqrt(n*p*(1-p))
## [1] 0.8485281
- 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 distribution video
dpois(0,n*p)
## [1] 0.449329
Expected value:
n*p
## [1] 0.8
Standard deviation:
- This is the square root of the mean (n*p)
(n*p)^0.5
## [1] 0.8944272