1. The Pareto distribution has two defining parameters, å (>0) and m (the minimum value of x). The å or α is called the Pareto index while m is the scale parameter.
P(X>x)) = (m/x) ^ å for all x ≥ m
Find suitable expressions for the pdf, the cumulative function, the mean, median, mode.
Special cases: m=1, m=σ
2. Let Y = log (X/m) where X is a Pareto distribution. Show that Y is exponentially distributed.
3. The Gamma distribution is another two-parameter distribution X~Gamma (α,β) also written Γ(α,β) . Both α and β are assumed positive.
too hard for even FM?
1. Pareto pdf åm^å / x^(å-1); CDF 1-(m/x)^å µ=åm/(å-1) median = m (2)^(1/å) mode=m
2. P(Y<y) = P(log X/m <y) = P(X<me^y) = 1- (m / me^y) ^å = 1 - e^(-åy)
