Worksheet Problem 1

Plot of the gamma pdf for part (a)

Plot[PDF[GammaDistribution[2,18],x],{x,0,150},PlotRange->All]

Probability for part (b)

CDF[GammaDistribution[2,18],30]//N

Worksheet Problem 3

Part (a): cumulative probabilities for X~Gamma(10,1)

gammaVals=Table[CDF[GammaDistribution[10,1],x],{x,1,20}]//N

Part (b): cumulative probabilities for Z~Normal(10,(Sqrt[10]))

normalVals=Table[CDF[NormalDistribution[10,(Sqrt[10])],x],{x,1,20}]//N

Comparison plot:

ListPlot[{gammaVals,normalVals},PlotLegends->{"Gamma","Normal"}]

Parts (c) and (d): cumulative probabilities for X~Gamma(100,1) and Z~Normal(100,10)

gammaVals=Table[CDF[GammaDistribution[100,1],x],{x,50,150,5}]//N
normalVals=Table[CDF[NormalDistribution[100,(Sqrt[100])],x],{x,50,150,5}]//N
ListPlot[{gammaVals,normalVals},PlotLegends->{"Gamma","Normal"},DataRange->{50,150}]

The gamma distribution converges to the normal distribution as α->∞

Manipulate[
Plot[PDF[GammaDistribution[α,1],x],{x,0,100},PlotRange->{0,0.2}],
{α,1,100}]