Here is another infectious disease model. Once a person becomes infected, the time X, in days,...

f(x,α,β) = (α/β^α)x^α−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

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