Let X a, denote a r.v. having the -distribution with parameters a and . (a) What is

Let X_a,ν denote a r.v. having the Γ-distribution with parameters a and ν.

(a) What is the difference between P(X_1,ν < x) and P(aX_a,ν < x)?

(b) For which ν does the Γ-density f_1,ν(x) converges to ∞ as x →0? For which ν is this limit equal to zero? Are there other possibilities? What about lim_x→∞ f_1,ν(x)? Which of the answers above will be different for f_a,ν(x)?

(c) Show that f_1,ν+1(x) ≤ f_1,ν(x) for x ≤ ν, and f_1,ν+1(x) ≥ f_1,ν(x) for x ≥ ν. Sketch the graphs of both functions for ν = 2 in the same picture.

(d) Using Excel or another software, find the probability that X_2,16 will fall within one standard deviation from its expected value.