1.
TP(T=t)−30.2750.470.33
Check the sum of probabilities
0.27+0.4+0.33=1
The expected value is
E[T]=−3⋅0.27+5⋅0.4+7⋅0.33=3.5
The variance is
Var[T]=E[T2]−E2[T]= ((−3)2⋅0.27+52⋅0.4+72⋅0.33)−(3.5)2= 16.35
The standard deviation is
σ[T]=Var[T]=16.35≈4.04
2.
XP(X=x)00.2110.4420.0630.1140.18
Check the sum of probabilities
0.21+0.44+0.06+0.11+0.18=1
The expected value is
E[X]= 0⋅0.21+1⋅0.44+2⋅0.06+3⋅0.11+4⋅0.18= 1.61
The variance is
Var[X]= (02⋅0.21+12⋅0.44+22⋅0.06+32⋅0.11+42⋅0.18)−(1.61)2≈1.96
The standard deviation is
σ[X]=1.96≈1.40
3.
YP(Y=y)270.3300.4140.15430.15
Check the sum of probabilities
0.3+0.4+0.15+0.15=1
The expected value is
E[Y]=27⋅0.3+30⋅0.4+14⋅0.15+43⋅0.15= 28.65
The variance is
Var[Y]= (272⋅0.3+302⋅0.4+142⋅0.15+432⋅0.15)−(28.65)2≈64.63
The standard deviation is
σ[Y]=64.63≈8.04
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