Help getting started, Linear Combinations and Independent Random Variables

thenewguy93

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I'm new here, so if I'm doing something wrong please help me to correct it. Now, I have no idea how to start working on this one. I have these equations in my notes: Mean- a1*u1+b*u2, Variance- a^2*o1^1+b^2*o2^2, Standard deviation- (Varience)^(1/2)
But I have no idea how to work with L and W :confused:
 

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I'm new here, so if I'm doing something wrong please help me to correct it. Now, I have no idea how to start working on this one. I have these equations in my notes: Mean- a1*u1+b*u2, Variance- a^2*o1^1+b^2*o2^2, Standard deviation- (Varience)^(1/2)
But I have no idea how to work with L and W :confused:
Let me re-write your two equations, which tell you "all you need to know" about propagation of errors for a linear combination \(\displaystyle a\ x_1 + b\ x_2\):

For independent random variables,

\(\displaystyle E[a\ x_1 + b\ x_2] = a\ E[x_1] + b\ E[x_2] = a\ \mu_1 + b\ \mu_2\)

\(\displaystyle V[a\ x_1 + b\ x_2] = a^2\ V[x_1] + b^2\ V[x_2] = a^2\ \sigma_1^2 + b^2\ \sigma_2^2\)

MEMORIZE THOSE TWO RULES ! !

ok - lets look at the specific questions.

Given: \(\displaystyle E[x_1] = \mu_1 = 75 \;\;\;\; V[x_1] = \sigma_1^2 = 256\)
.........\(\displaystyle E[x_2] = \mu_2 = 50 \;\;\;\; V[x_2] = \sigma_2^2 = 81\)

a) \(\displaystyle L = 4 + 3\ x_1\)
The constant has no Variance. It must be included in the \(\displaystyle E[L]\), but does not contribute to \(\displaystyle V[L]\).

......\(\displaystyle E[L] = 4 + 3\ E[x_1] = \ \cdot\ \cdot\ \cdot\)

......\(\displaystyle V[L] = 3^2\ V[x_1] = \ \cdot\ \cdot\ \cdot\)

......\(\displaystyle \sigma_L = \sqrt{V[L]} = \ \cdot\ \cdot\ \cdot\)

b) \(\displaystyle W = x_1 - x_2\)

......\(\displaystyle E[W] = E[x_1] + (-1)\ E[x_2] = \ \cdot\ \cdot\ \cdot\)

......\(\displaystyle V[W] = V[x_1] + (-1)^2\ V[x_2] = \ \cdot\ \cdot\ \cdot\)

......\(\displaystyle \sigma_W = \sqrt{V[W]} = \ \cdot\ \cdot\ \cdot\)

NOTE that the coefficients a and b are squared, so Variances always add.

If you need more help, show us your work.
 
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