(*  hw 3 prob 3-10    10-11-23   *)



(*   part a    tk = k h   h=1/m   *)

ise$k = Expand[ Integrate[ (vk/(n (1/m) )-1/m)^2, {x,k/m, (k+1)/m} ] ]

mise$k = (     ( ise$k /. vk^2 -> n pk (1-pk) + (n pk)^2 ) )


(*   hw 3   problem 4     beta   lognormal   *)


fx$beta = Gamma[a+b] / ( Gamma[a] Gamma[b] ) x^(a-1) (1-x)^(b-1)

m = Integrate[ x fx$beta, {x, 0, 1}, Assumptions -> {a > 1, b > 1}]
v = Integrate[ (x-m)^2 fx$beta, {x, 0, 1}, Assumptions -> {a > 1, b > 1}]

rfb = Integrate[ D[ fx$beta,x ]^2, {x, 0, 1}, Assumptions -> {a > 2, b > 2}]

cons = ( 6/rfb )^(1/3) / Sqrt[v]    (*  hstar = cons * Sqrt[v] * n^(-1/3)  *)

log$cons = PowerExpand[ Log[ ( cons /. b -> a ) ] ]

LogPlot[ { Exp[log$cons], 3.49 }, { a, 4,750} ]



(*   lognormal   *)


fxl = Exp[ -Log[x]^2/2 ] / ( x Sqrt[2Pi] )

Integrate[ fxl, {x,0,Infinity} ]
m = Integrate[ x fxl, {x,0,Infinity} ]
v = Integrate[ (x-m)^2 fxl, {x,0,Infinity} ]

rfb = Integrate[ D[ fxl,x ]^2, {x, 0, Infinity} ]

cons$ln = (6/rfb)^(1/3) / Sqrt[v]

N[ cons$ln ]