Suppose X and Y are 2 independent normal random variable such that x~N(65,28) and Y~N(85,36).Find p(134<X+Y<166) and p(Y>x)
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Expert's answer
2021-08-19T15:20:32-0400
The distribution of a sum of two normally distributed independent variables X∼N(μ1,σ12) and Y∼N(μ2,σ22) is another normal distribution (X+Y)∼N(μ1+μ2,σ12+σ22).
Given X∼N(65,28),Y∼N(85,36). Then (X+Y)∼N(65+85,28+36).
P(134<X+Y<166)
=P(X+Y<166)−P(X+Y≤134)
=P(Z<64166−150)−P(Z≤64134−150)
=P(Z<2)−P(Z≤−2)
≈0.97725−0.02275=0.9545
The distribution of a difference of two normally distributed independent variables Y∼N(μ2,σ22) and X∼N(μ1,σ12) is another normal distribution (Y−X)∼N(μ2−μ1,σ12+σ22).
Given X∼N(65,28),Y∼N(85,36). Then (Y−X)∼N(85−65,28+36).
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