Calculating Expectation Values for Independent Random Variables

[toothpaste666]

Homework Statement

If X1 has mean -3 and variance 2 while X2 has mean 5 and variance 4 and the two are independent find
a) E(X1 - X2)
b) Var(X1 - X2)

The Attempt at a Solution

I am not very clear on what I am supposed to be doing for this problem. I don't fully understand this expectation value concept. Can someone help explain what they are asking me to do or give me a hint on how to get started please? Here is what I did so far but I don't know if its worth anything

they are independent so my book says the covariance is 0
so
E[(X1 - μ1)(X2 - μ2)] = 0
E[(X1 + 3)(X2 - 5)] = 0
E[X1X2 - 5X1 + 3X2 -15] = 0

I think that is equal to this but I may be wrong:

E[X1X2] - 5E[X1] + 3 E[X2] - 15 = 0

I don't know what to do from here though. Am I on the right track at all?

 

[andrewkirk]

Yes you are on the right track.

'The mean of X1' is defined to be E[X1].

As regards E[X1X2]: have they given you the rule for the expectation value of a product of independent variables?
If not, it's easy to derive. Start by writing the expectation as a double integral.

 

[LCKurtz]

Homework Statement

If X1 has mean -3 and variance 2 while X2 has mean 5 and variance 4 and the two are independent find
a) E(X1 - X2)
b) Var(X1 - X2)

The Attempt at a Solution

I am not very clear on what I am supposed to be doing for this problem. I don't fully understand this expectation value concept. Can someone help explain what they are asking me to do or give me a hint on how to get started please? Here is what I did so far but I don't know if its worth anything

they are independent so my book says the covariance is 0
so
E[(X1 - μ1)(X2 - μ2)] = 0
E[(X1 + 3)(X2 - 5)] = 0
E[X1X2 - 5X1 + 3X2 -15] = 0

I think that is equal to this but I may be wrong:

E[X1X2] - 5E[X1] + 3 E[X2] - 15 = 0

I don't know what to do from here though. Am I on the right track at all?

No, I don't think you are on the right track. You are making it much too difficult. You don't need to do any calculations with covariance or the expectation of a product. Surely your text has formulas for the expected value and variance of a sum of independent random variables. Have you looked for such formulas?

 

[toothpaste666]

Ok sorry my book is very confusingly written but I think I figured it out.
so E(aX1 + bX2) = aE(X1) + bE(X2)
for part a) a = 1 and b = -1
so E(X1-X2) = E(X1) - E(X2) = -3 - 5 = -8

for part b)
Var(aX1+bX2) = a^2Var(X1) + b^2Var(X2)
since a =1 and b = -1
Var(X1-X2) = Var(X1) + Var(X2) = 2 + 4 = 6

is this correct?

 

[andrewkirk]

Yes that's correct. Just be careful not to forget that the formula you use in (b) only holds when the two random variables are independent. The formula in (a) holds regardless of dependence.

 

[toothpaste666]

thank you!