Linear Combinations (Probability) question

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• Sep 28th 2008, 06:35 PM
agbrianlee355
Linear Combinations (Probability) question
I would like to start by thanking mrfantastic for solving BOTH my prob. questions (Bow). Thank you!!

Anywayz, heres my question:

You know the rule for Linear Combinations where the Variance of say, 2X + 3Y is something like Var(2X+3Y) = 2^2Var(X) + 3^2Var(Y)? But how come on some questions I came across you don't square the co-efficient (2 and 3) ?
(i.e Var (2X+3Y) = 2Var(X) + 3Var(Y) )

In other words, what conditions must be met in order to square it or not? (please give some examples)

Thanks in advance! =D
• Sep 29th 2008, 03:03 AM
mr fantastic
Quote:

Originally Posted by agbrianlee355
I would like to start by thanking mrfantastic for solving BOTH my prob. questions (Bow). Thank you!!

Anywayz, heres my question:

You know the rule for Linear Combinations where the Variance of say, 2X + 3Y is something like Var(2X+3Y) = 2^2Var(X) + 3^2Var(Y)? But how come on some questions I came across you don't square the co-efficient (2 and 3) ? Mr F says: What questions?

(i.e Var (2X+3Y) = 2Var(X) + 3Var(Y) ) Mr F says: This is wrong.

In other words, what conditions must be met in order to square it or not? (please give some examples)

Thanks in advance! =D

If X and Y are INDEPENDENT random variables, then $\text{Var} (aX + bY) = a^2 \, \text{Var} (X) + b^2 \, \text{Var} (Y)$. $\text{Var} (aX + bY) = a \, \text{Var} (X) + b \, \text{Var} (Y)$ is false (unless a = b = 1). There are NO conditions (except a = b = 1) under which $\text{Var} (aX + bY) = a \, \text{Var} (X) + b \, \text{Var} (Y)$.

If X and Y are NOT INDEPENDENT random variables, then $\text{Var} (aX + bY) = a^2 \, \text{Var} (X) + b^2 \, \text{Var} (Y) + 2ab \, \text{Cov} (X, Y)$ where Cov(X, Y) is the covariance of X and Y.

Note: If X and Y are independent random variables then Cov(X, Y) = 0.
• Sep 29th 2008, 12:04 PM
agbrianlee355
An example of this type of question=
----

A beam of wood consists of three pieces of wood of type A, plus two pieces of wood of type B.

The thickness of A has mean 2mm and variance 0.04mm^2.
The thickness of B has mean 1mm and variance 0.01mm^2.

Find the mean and variance of the beam of wood.

----

The answers shown are=
Mean = E(3A+2B) = 3E(A) + 2E(B) = 8mm
Variance = Var(3A+2B) = 3Var(A) + 2Var(B) = 0.14mm^2.

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(btw this is from an A level CIE textbook)
• Sep 29th 2008, 12:53 PM
mr fantastic
Quote:

Originally Posted by agbrianlee355
An example of this type of question=
----

A beam of wood consists of three pieces of wood of type A, plus two pieces of wood of type B.

The thickness of A has mean 2mm and variance 0.04mm^2.
The thickness of B has mean 1mm and variance 0.01mm^2.

Find the mean and variance of the beam of wood.

----

The answers shown are=
Mean = E(3A+2B) = 3E(A) + 2E(B) = 8mm
Variance = Var(3A+2B) = 3Var(A) + 2Var(B) = 0.14mm^2. Mr F says: The answer is wrong. It should be 0.4 mm^2.

----

(btw this is from an A level CIE textbook)

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