In: Statistics and Probability
Here is another infectious disease model. Once a person becomes
infected, the time X, in days, until the person becomes infectious
(can pass on the disease) can be modeled as a Weibull random
variable with density function
f(x,α,β) = (α/βα)xα−1e−(x/β)α
for 0 ≤ x ≤ ∞ and 0 otherwise with α = 3.7 and
β = 7.1
α is the shape parameter and β is the scale
parameter. Hint: Solve this with the built-in R functions for the
Weibull distribution (dweibull(),pweibull(), qweibull()) not f as
defined above. Otherwise you may get intermediate values too large
to use. For a) and b) the text (and notes) give formulas for the
answers. You can calculate from these formulas. Note that these
formulas use the gamma function. Γ(α) is the gamma
function. In R, there is a built-in function gamma() which
calculates this.
a) What is the expected value of X?
b) What is the variance of X?
c) What is the standard deviation of X?
d) What is the probability that X is larger than its expected
value?
e) What is the probability that X is > 2?
f) What is the probability that X is > 4?
g) What is the probability that X > 4 given that X > 2?
h) Calculate the 70th percentile of X