Example 20.4 on p. 264 of Billingsley's book "Probability and Measures"
The distribution of the length of the maximum subinterval generated by i.i.d. n uniform random variables on [0,1]
By Le Chen.
Crated on Thu 19 Jan 2023 07:37:53 AM CST
The distribution of the length of the maximum subinterval generated by i.i.d. n uniform random variables on [0,1]
By Le Chen.
Crated on Thu 19 Jan 2023 07:37:53 AM CST
First give the cumulative distribution function
First give the cumulative distribution function
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ψ[x_,n_]:=Sum[Binomial[n+1,k],{k,0,n+1}]
k
(-1)
n
Max[1-kx,0]
Plots of the CDF' s
Plots of the CDF' s
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Plot[Table[ψ[x,k],{k,1,6}],{x,0,1}]
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Plots of CDF and PDF for different n
Plots of CDF and PDF for different n
n = 1
n = 1
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n=1;D[ψ[x,n],x]//FullSimplifyPlot[{%,ψ[x,n]},{x,0,1}]Clear[n]
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n = 2
n = 2
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n=2;D[ψ[x,n],x]//FullSimplifyPlot[{%,ψ[x,n]},{x,0,1}]Clear[n]
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n = 3
n = 3
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n=3;D[ψ[x,n],x]//FullSimplifyPlot[{%,ψ[x,n]},{x,0,1}]Clear[n]
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n = 4
n = 4
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n=4;D[ψ[x,n],x]//FullSimplifyPlot[{%,ψ[x,n]},{x,0,1},PlotRangeFull]Clear[n]
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n = 5
n = 5
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n=5;D[ψ[x,n],x]//FullSimplifyPlot[{%,ψ[x,n]},{x,0,1},PlotRangeFull]Clear[n]
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