SOLUTION: A random variable X has mean 110 and standard deviation 18, and a random variable Y has mean 215 and standard deviation 30; and the correlation between X and Y is r=0.75. The stand

Algebra ->  Probability-and-statistics -> SOLUTION: A random variable X has mean 110 and standard deviation 18, and a random variable Y has mean 215 and standard deviation 30; and the correlation between X and Y is r=0.75. The stand      Log On


   

Question 503214: A random variable X has mean 110 and standard deviation 18, and a random variable Y has mean 215 and standard deviation 30; and the correlation between X and Y is r=0.75. The standard deviation of X+Y is [x]. (Round your answer to 2 decimal places.)
Answer by Sarpi(32) About Me  (Show Source):
You can put this solution on YOUR website!
E(.)= the mean
s(.)= the standard deviation
So, E(X)=110 s(X)=18 ; E(Y)=215 s(Y)=30
r(X,Y)=0.75
The standard deviation of X+Y, formula;
Var%28X%2BY%29=+Var%28x%29%2BVar%28y%29%2B2cov%28x%2Cy%29 "Var= s%5E2 "
The above formula comes from this general formula
Var%28aX%2BbY%29=+a%5E2Var%28x%29%2Bb%5E2Var%28y%29%2B2abcov%28x%2Cy%29 note: a and b are equal to 1, the coefficient of X and Y
Var%28X%2BY%29=+18%5E2%2B30%5E2%2B2cov%28x%2Cy%29. We therefore need to find the covariance of X and Y
Formula: r%28x%2Cy%29=+Cov%28x%2Cy%29%2F%28s%28x%29%2As%28y%29%29 . The actual formula for finding the cov(x,y) is E%28XY%29-E%28X%29%2AE%28Y%29. We don't know the value of E(XY) so we use the alternative since r(x,y) is given.
r%28x%2Cy%29=+Cov%28x%2Cy%29%2F%28s%28x%29%2As%28y%29%29
0.75=+Cov%28x%2Cy%29%2F%2818%2A30%29
Cov%28x%2Cy%29=+0.75%2A%2818%2A30%29
Cov(x,y)=405
However, Var%28X%2BY%29=+18%5E2%2B30%5E2%2B2cov%28x%2Cy%29
Var%28X%2BY%29=+18%5E2%2B30%5E2%2B2%2A405
Var%28X%2BY%29=+2034
That implies sqrt%282034%29 is equal to the standard deviation of X+Y
So, s(X+Y)=45.10
I hope you really understand the steps. And your feedback will be appreciated.