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Math Help - Random Functions

  1. #1
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    Random Functions

    At a heat-treating company, iron castings and steel forgings are heat-treated to achieve desired mechanical properties and machinability. One steel forging is annealed to soften the part for each machining. Two lots of this part, made of 1020 steel, are heat-treated in two different furnaces. The specification for this part is 36-66 on the Rockwell G scale. Let X1 and X2 equal the respective hardness measurements for parts selected randomly from furnaces 1 and 2. Assume that the distributions of X1 and X2 are N(47.88, 2.19) and N(43.04, 14.89), respectively.

    What are the p.d.f.s of X1 and X2?
    Compute P(X1>X2), assuming independence of X1 and X2.

    Thank you for your help!
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  2. #2
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    Quote Originally Posted by calabrone View Post
    At a heat-treating company, iron castings and steel forgings are heat-treated to achieve desired mechanical properties and machinability. One steel forging is annealed to soften the part for each machining. Two lots of this part, made of 1020 steel, are heat-treated in two different furnaces. The specification for this part is 36-66 on the Rockwell G scale. Let X1 and X2 equal the respective hardness measurements for parts selected randomly from furnaces 1 and 2. Assume that the distributions of X1 and X2 are N(47.88, 2.19) and N(43.04, 14.89), respectively.

    What are the p.d.f.s of X1 and X2?
    Compute P(X1>X2), assuming independence of X1 and X2.

    Thank you for your help!
    You are given that the distribution of X1 is N(47.88,2.19), that is it is normal with mean 47.88 and variance 2.19. Look up the pdf of the normal distribution.

    Same procedure for X2.

    X3 = X1-X2 ~N(47.88-43.04, 2.19+14.89)

    So P(X1>X2)=P(X3>0).

    CB
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  3. #3
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    I'm not sure where to find the pdfs of the normal distribution.

    Can I get the answer to the second part from the normal distribution table? I looked on the P(Z>z) table to find P(X3>0)=0.5000

    Thank you
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  4. #4
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    Quote Originally Posted by calabrone View Post
    I'm not sure where to find the pdfs of the normal distribution.
    wikipedia: normal distribution

    Can I get the answer to the second part from the normal distribution table? I looked on the P(Z>z) table to find P(X3>0)=0.5000

    Thank you
    You compute the z-score corresponding to X3=0 when X3~N(47.88-43.04, 2.19+14.89), (and remember that the second argument here is the variance not the SD). Now look up the z-score in the normal table.

    CB

    CB
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    Would you be able to show me the work for this problem or one similar so I can follow your example to solve my remaining problems? Thank you for all your help.
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  6. #6
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    Quote Originally Posted by CaptainBlack View Post
    wikipedia: normal distribution



    You compute the z-score corresponding to X3=0 when X3~N(47.88-43.04, 2.19+14.89), (and remember that the second argument here is the variance not the SD). Now look up the z-score in the normal table.
    The z-score corresponding to x_3=0, when X_3 \sim N(47.88-43.04, 2.19+14.89)=N(4.84,17.18) is:

    z=\frac{x_3-4.84}{\sqrt{17.18}}=\frac{0-4.84}{\sqrt{17.18}}=-1.16771

    Now we look -1.16771 up in the normal table or using a normal distribution calculator (available on line here) to get the required probability of \approx 0.1215

    CB
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  7. #7
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    Thank you so much. For the first part, I still do not know what chart I should get the pdfs from. I attached the file of my work showing a graph of the pdfs, but I don't know how to evaluate their equations to get a normal pdf. Thank you again.
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