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Math Help - Random Variable (Normal Distribution)

  1. #1
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    Random Variable (Normal Distribution)

    Hello,

    Struggling to answer the following question:

    The random variable Z has the mean -4 and standard deviation 5. Find:
    E[3-2Z]
    E[(3-2Z)^2]

    Now, I think this is a normal distribution, Z~N(-4, 5^2).

    And that E(Z) = -4, but as it has more terms in the E[], i'm a little confused on what to do!

    Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by MattWT View Post
    Hello,

    Struggling to answer the following question:

    The random variable Z has the mean -4 and standard deviation 5. Find:
    E[3-2Z]
    E[(3-2Z)^2]

    Now, I think this is a normal distribution, Z~N(-4, 5^2).

    And that E(Z) = -4, but as it has more terms in the E[], i'm a little confused on what to do!

    Thanks
    By properties of expected value, E[3-2Z] = 3-2E[Z].

    Now with regards to E[(3-2Z)^2], there's a quick way to do this using the definition of variances:

    \text{Var}[3-2Z]+(E[3-2Z])^2=E[(3-2Z)^2]

    But then by properties of variances, \text{Var}[3-2Z] = \text{Var}[-2Z]. Thus, we can write this as:

    E[(3-2Z)^2]=4\text{Var}[Z]+(E[3-2Z])^2.

    Once you find E[3-2Z], then you should be able to find E[(3-2Z)^2] because we happen to know what \text{Var}[Z] is.

    Can you proceed?
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