# How can I plot this function in Mathematica for different values of $a, b$ [closed]

Prem[a_, b_] :=
Integrate[(Gamma[a + b]/(Gamma[a]*Gamma[b]))*
Integrate[
Integrate[u^(a - 1)*(1 - u)^(b - 1), {u, 0, x}], {x, 0,
tailG[x]}], {x, 0, Infinity}];
data = {Prem[0.5, 1], Prem[0.6, 1], Prem[0.7, 1], Prem[0.8, 1],
Prem[0.9, 1], Prem[1, 1], Prem[0.5, 2], Prem[0.6, 2], Prem[0.7, 2],
Prem[0.8, 2], Prem[0.9, 2], Prem[1, 2], Prem[0.5, 3], Prem[0.6, 3],
Prem[0.7, 3], Prem[0.8, 3], Prem[0.9, 3], Prem[1, 3], Prem[0.5, 4],
Prem[0.6, 4], Prem[0.7, 4], Prem[0.8, 4], Prem[0.9, 4],
Prem[1, 4], Prem[0.5, 5], Prem[0.6, 5], Prem[0.7, 5], Prem[0.8, 5],
Prem[0.9, 5], Prem[1, 5]}

ListLinePlot[data]

Plot[{Prem[a, b],Prem1[a,b]},Prem2[a,b]}, {x, First[data][[1]], Last[data][[1]]},
PlotRange -> Full]


for a<1 and b>1 i want to plot and then include more that two functions in the some plot

• What is tailG? May 17 at 11:00
• tailG[x_] := Exp[-(1/2)*x]; is survival function May 17 at 11:04
• what is Prem[]? May 17 at 12:38
• its my function May 17 at 20:14

As I understand it, you want to plot that 3D-integral depending on the parameters a,b. Let us do it step by step (Two small steps are better than one big step.). First, we calculate

Gamma[a + b]/(Gamma[a]*Gamma[b])*Integrate[u^(a - 1)*(1 - u)^(b - 1), {u, 0, x},
Assumptions -> a > 0 && a < 1 && b > 1 && x >= 0]


ConditionalExpression[(Beta[x, a, b]*Gamma[a + b])/(Gamma[a]*Gamma[b]), x < 1]

Second, we make the next integration (The upper bound of the integration is redesignated from Exp[-x/2] to Exp[-y/2] to avoid any misunderstanding.)

Integrate[(Beta[x, a, b] Gamma[a + b])/(Gamma[a] Gamma[b]), {x, 0, Exp[-y/2]},
Assumptions -> a > 0 && a < 1 && b > 1 && y >= 0]


ConditionalExpression[(Gamma[a + b]*Hypergeometric2F1Regularized[a, 1 - b, 2 + a, E^(-1/2*y)])/ (E^(((1 + a)*y)/2)*Gamma[b]), && y > 0]

In the above Mathematica blows on cold water by NotElement[b, Integers] (see that follows). It's clear there is no chance to symbolically integrate the latest expression by y, so

Plot3D[Evaluate[NIntegrate[(Gamma[a + b]* Hypergeometric2F1Regularized[a, 1 - b, 2 + a,
E^(-1/2*y)])/(E^(((1 + a)*y)/2)*Gamma[b]), {y, 0, Infinity},
PrecisionGoal -> 5, AccuracyGoal -> 5]], {a, 0, 1}, {b, 1, 5}]


• Excuse me but in my mathematica doesnt run this graph May 18 at 10:23
• I use 12.2 on Windows 10 Pro. Here is the executed code exported as PDF. .nb file on demand through Dropbox. Don't hesitate to ask for further explanation in need. May 18 at 13:40