- 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
\[ P(X=1)= \frac {kn−(k−1)^n}{kn}\]
\[ P(X=2)=\frac{(k−2+1)n−(k−2)^n}{kn}\]
\[P(X=y)=\frac{(k−y+1)^n−(k−y)n}{k^n}\]
- 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.).
- 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..)
pgeom(8, 0.1, lower.tail=FALSE)
## [1] 0.3874205
Expected \[ E(X)=q/p=0.9/0.1=9 \]
Standard deviation \[ σ2= \sqrt {q/p^2} =\sqrt {0.9/0.1^2} ≈ 9.4868.\]
- What is the probability that the machine will fail after 8 years?. Provide also the expected value and standard deviation. Model as an exponential.
pexp(8, 0.1, lower.tail = F)
## [1] 0.449329
Expected value is \[ E(X)=1/λ=1/0.1=10 \]
Standard deviation \[ σ2= \sqrt {1/λ^2} = 1/λ=10 \]
- 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)
pbinom(0, 8, 0.1)
## [1] 0.4304672
Expected value \[ E(X)=np=8×0.1=0.8 \]
Standard deviation \[σ2=\sqrt {npq}= \sqrt{8×0.1×0.9} ≈0.8485.\]
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.
ppois(0,0.1,lower.tail=TRUE)^8
## [1] 0.449329
For Poisson distribution, E(X)=σ2=λ=0.1