Other distributions

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)

 However, © David Scoins 2017