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Thread: Expected Value & Variance from Z = min (X,Y)

  1. #1
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    Expected Value & Variance from Z = min (X,Y)

    Dear all,

    I have an assignment, but I don't understand yet which formulas I have to use to solve it. Here is the question:

    There is an equipment that consist of 2 components. Each of them has a "life time" X and Y. X and Y are independent, and here are the distribution functions:

    That equipment will be damaged if one of the components is damaged. The "life time" for that equipment is . Please calculate the E(Z) and Var(Z)!

    I have seen some examples and explanations from the other sources, but it difficult for me to understand because I study lonely without a lecturer.
    I think it is the formula that I have to use:

    Please explain and guide me the steps to solve it. Thanks for your help.
    Last edited by statjunior; 09-24-2014 at 01:26 AM.

  2. #2
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    Re: Expected Value & Variance from Z = min (X,Y)

    This is a very classical question. Here is my two cents:

    1. The formula you post is the general formula:

    E[g(X, Y)] = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} g(x,y)f_{X,Y}(x,y)dxdy

    The problem is you need to re-write the minimum function as a piecewise function first and thus you can simplify the integral into the form you can integrate:

    \min(x,y) = \begin{cases} x ~~~~\text{if}~~ x \leq y \\ y ~~~~\text{if}~~ y < x \end{cases}

    That is, the integral will be splited into two parts; Since now x is the variable of the inner integral, it will be splited like this:

    E[\min(X, Y)]

    = \int_{-\infty}^{+\infty}\int_{-\infty}^{y} \min(x,y)f_{X,Y}(x,y)dxdy 
+ \int_{-\infty}^{+\infty}\int_{y}^{+\infty} \min(x,y)f_{X,Y}(x,y)dxdy

    = \int_{-\infty}^{+\infty}\int_{-\infty}^{y} xf_{X,Y}(x,y)dxdy 
+ \int_{-\infty}^{+\infty}\int_{y}^{+\infty} yf_{X,Y}(x,y)dxdy

    and you may proceed from here.

    I assume you know the joint pdf here as each of them are just exponential random variables and they are independent.

    2. However, the main spirit of this question is want you to figure out the following relation:

    \min(x, y) \geq z \iff x \geq z \text { and } y \geq z

    Try to understand and use this relationship to calculate the CDF of \min(X, Y):

    F_Z(z) = \Pr\{\min(X, Y) \leq z\}

    Once you find out the CDF you will know that actually Z is just another exponential random variables and of course the mean and variance are easy to find out.

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