Integrate[ c E^(-(2x+3y)),{x,0,Infinity},{y,0,Infinity}]

c==6;

Integrate[6 E^(-(2x+3y)),{x,0,1/2},{y,0,1/3}]//N

EX=0

Maximize[{t^3*(1-2t)^6,0<t<1,0<1-2t<1},t]

1/6//N

Covariance[  ProbabilityDistribution[Which[
Abs[y]<x && 0<x<1, 1,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]

Probability[Max[x,y]<1/2,{x,y} \[Distributed] ProbabilityDistribution[Which[
 0<x<y<1,6x,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]//N

Probability[Max[x,y]<1/2,{x,y} \[Distributed] ProbabilityDistribution[Which[
 0<x<y<1,6x,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]//N

Expectation[1/x^2,{x} \[Distributed] ProbabilityDistribution[Which[
x>0,2x^3 E^(-x^2),True,0
],{x,-Infinity,Infinity}]]-Expectation[1/x,{x} \[Distributed] ProbabilityDistribution[Which[
x>0,2x^3 E^(-x^2),True,0
],{x,-Infinity,Infinity}]]^2//N

Expectation[X1+X2+X3,{X1\[Distributed]BinomialDistribution[1,1/6],X2\[Distributed]BinomialDistribution[2,1/6],X3\[Distributed]BinomialDistribution[3,1/6]
}
]

Probability[X1+X2+X3==1,{X1\[Distributed]BinomialDistribution[1,1/6],X2\[Distributed]BinomialDistribution[2,1/6],X3\[Distributed]BinomialDistribution[3,1/6]
}
]//N

Area@ImplicitRegion[ (X>1 || Y>Sqrt[3]/2)&&X^2+Y^2≤1,{X,Y}]/Pi//N

Integrate[Sin[x],{x,0,Pi},{y,0,1}]

TransformedDistribution[x+1/2 y,{x,y}\[Distributed] ProbabilityDistribution[Which[
0<x<1&& 0<y<2x,2x y,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]//PDF[#,z]&//FullSimplify

ProbabilityDistribution[Which[
0<x<1&& 0<y<2x,2x y,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]//Covariance

Expectation[(x+1/2 y)^2,{x,y}\[Distributed] ProbabilityDistribution[Which[
0<x<1&& 0<y<2x,2x y,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]-Expectation[x+1/2 y,{x,y}\[Distributed] ProbabilityDistribution[Which[
0<x<1&& 0<y<2x,2x y,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]]^2

MarginalDistribution[  ProbabilityDistribution[Which[
0<x<1&& 0<y<2x,2x y,True,0
],{x,-Infinity,Infinity},{y,-Infinity,Infinity}],1]//PDF[#,x]&

Integrate[ 2 x y,{y,0,2x}]

(\[Placeholder]/(10^4))

InverseCDF[  StudentTDistribution[5],0.85]

Probability[ x<InverseCDF[  StudentTDistribution[5],85/100]^2,x\[Distributed]FRatioDistribution[1,5]]//N

Mean[ x-y]

Expectation[ 1/x,x\[Distributed] UniformDistribution[{0,1}]]

TransformedDistribution[ X+Y,{X \[Distributed] NormalDistribution[1,2],Y\[Distributed] NormalDistribution[2,3]}]

FRatioDistribution[1,6]

1-((10^4-1)/(10^4))*((10^4-2)/(10^4))*((10^4-3)/(10^4))*((10^4-4)/(10^4))*((10^4-5)/(10^4))*((10^4-6)/(10^4))//N

Variance[ExponentialDistribution[4]]

Solve[1/4 ==(x/(2 *(1/4)))]

4+1/16 -2*1/8//#/36&//N

Variance[BinomialDistribution[600,6/10]]

(4+1/16-2*1/8)/36//N

Mean[ExponentialDistribution[4]]

RegionPlot[ x^2+y^2≤1,]

ImplicitRegion[x^2+y^2≤1&& (x>1/2 || y>Sqrt[3]/2),{x,y}]//RegionPlot

1.96^2 * 4

FRatioDistribution[1,6]

PDF[FRatioDistribution[1,6],x]//FunctionExpand

SurvivalDistribution

SurvivalFunction[ExponentialDistribution[1/5],5]

TransformedDistribution[Sqrt[X],X\[Distributed] PoissonDistribution[1]]