Q3)

v0=(Sqrt[((2*1.3806488*(10^(-23))*300)/(28.02*1.660538921*(10^(-27))))]);
MD[v_]:=(4/(Sqrt[π]))*((v^2)/(v0^3))*Exp[((-(v^2))/(v0^2))];
Plot[MD[v],{v,0,1200},PlotTheme->"Scientific",FrameLabel->{"v","D(v)
"},Prolog->{Red,Line[{{422,0},{422,1}}],Blue,Line[{{517,0},{517,1}}],Green,Line[{{476,0},{476,1}}]},PlotLabel->"Maxwell Distribution",PlotLegends->Placed[LineLegend[{Red,Blue,Green},{"v_max","v_rms","v_avg"},LegendLabel->"Speed Locations",LegendLayout->{"Column",2}],{0.8,0.7}]
]

Integrate[MD[v],{v,300,572}]

Q5)

a) & b)

mq1[q_]:=((Binomial[200+200-q,200-q])/(Binomial[200+200,200]))

ListPlot[Table[mq1[q],{q,0,10}],PlotTheme->"Scientific",FrameLabel->{"q","Ω
"},DataRange->{0,10},PlotStyle->Red,PlotRange->All]

FindFit[Table[mq1[q],{q,0,10}], a*Exp[-q/b],{a,b},q]

Show[ListPlot[Table[mq1[q],{q,0,10}],PlotStyle->Red,PlotTheme->"Scientific",FrameLabel->{"q","Ω
"},DataRange->{0,10}],Plot[a Exp[-((q+1)/b)]/. {a->2.0038898158801857`,b->1.4392349384490526`},{q,-1,10},PlotStyle->Black]]

d)

mq2[q_]:=((Binomial[2000+200-q,2000-q])/(Binomial[2000+200,2000]))

ListPlot[Table[mq2[q],{q,0,100}],PlotTheme->"Scientific",FrameLabel->{"q","Ω
"},PlotStyle->Red,PlotRange->All]

FindFit[Table[mq2[q],{q,0,100}],a*Exp[((-q)/b)],{a,b},q]

Show[ListPlot[Table[mq2[q],{q,0,100}],PlotRange->All,PlotStyle->Red,PlotTheme->"Scientific",FrameLabel->{"q","Ω
"},DataRange->{0,100}],Plot[a Exp[-((q+1)/b)]/.{a->1.1016088444611236`,b->10.444575609529062`},{q,-10,100},PlotStyle->Black]]