Probability Distribution of two normally distributed independent random variables.

Question

If X~N(10,3) and Y~N(15,8), and if X and Y are independent, what is the probability distribution of

1. X +Y

2. X-Y

3. 3X

4. 4X + 5Y

If X and Y are positively correlated with a correlation coefficient of 0.6, recalculate the probability distributions in each case? How different from case a are the distributions?

Answer 1

Most of the formulas that Deanna provided are correct. That last line is incorrect though.

 

Var(aX + bY) = a² * Var(X) + b² * Var(Y) + 2ab*Cov(X, Y), but Cov(X, Y) is definitely not the correlation coefficient. Cov(X, Y) is the covariance between random variables X and Y, and it's related to the correlation coefficient as follows:

 

cov(X, Y) = r * σ_X * σ_Y, where r is the correlation coefficient and σ is the standard deviation of a random variable.

 

 

Answer 2

Daniel,

 

This might help:

E(aX+bY)=a*E(x)+b*E(y)

V(aX)=a^2*V(X)

If independent V(aX+bY)=a^2*V(x)+b^2*V(Y)

If correlated V(aX+bY)=a^2*V(x)+b^2*V(Y)+2ab*Cov(X,Y)

where the Cov(X,Y) is the correlation coefficient.