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Math Help - Linear Combinations (Probability) question

  1. #1
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    Linear Combinations (Probability) question

    I would like to start by thanking mrfantastic for solving BOTH my prob. questions . 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
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  2. #2
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    Quote Originally Posted by agbrianlee355 View Post
    I would like to start by thanking mrfantastic for solving BOTH my prob. questions . 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.
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  3. #3
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    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.

    ----

    (btw this is from an A level CIE textbook)
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  4. #4
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    Quote Originally Posted by agbrianlee355 View Post
    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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