Suppose that X is uniformly distributed on 03 Y is uniformly

Suppose that X is uniformly distributed on (0,3), Y is uniformly distributed on (0,5), and X and Y are independent. Determine the probability that 3X - 2Y is less than 5.

Solution

Given

X~U(0,3)

Y~U(0,5)

E(X)=3/2

E(Y)=5/2

V(X)=9/12

V(Y)=25/12

Define a new random variable U=3X-2Y

E(U)=3E(X)-2E(Y)

      =3*3/2-2*5/2

    =-0.5

V(U)=9V(U)+4V(U)

       =9*9/12+4*25/12

       =15.0833

P[3X-2Y<5]=P[U<5]

                     =P[U-E(U)/sqrt{Var(U)}]<5+0.5/sqrt{15.08}

                    = P[U-E(U)/sqrt{Var(U)}]<5.5/3.8837

                   =1.416

                  =NORMSDIST(1.416){using EXCEL function)

                  =0.9216

The probability that 3X-2Y is lessthan 5 is 0.9216