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Math Help - Random sample from a continuous type dist

  1. #1
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    Random sample from a continuous type dist

    Let X1,X2,,,,Xn be a random sample from a continuous type distribution
    a) find P(X1<=X2),P(X1<=X2,X1<=X3),...,P(X1<=Xi, i=2,3,...,n).

    (The answer for this is 1/n but I am not sure the reasoning behind of that.
    Can anybody explain this?)

    b) Suppose the sampling continues until X1 is no longer the smallest observation, (i.e., Xj < X1 <= Xi, i=2,3,...,j-1). Let Y equal the number of trials until X1 is no longer the smallest observation, (i.e., Y=j-1). Show that the distribution of Y is P(Y=y) = 1 / y(y+1), y=1,2,3,...
    (????????)

    c) Compute the mean and variance of Y if they exist.
    (?????????)
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  2. #2
    MHF Contributor matheagle's Avatar
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    Since 1=P(X>Y)+P(X<Y).
    I would think you just have to show that these probabilities are equal,
    giving you .5.

    As for (c), if P(Y=y) = 1 / y(y+1), y=1,2,3,..., then

    E(Y)=\sum_{y=1}^{\infty}{1\over y+1}=\sum_{n=2}^{\infty}{1\over n}=\infty.
    Thus the second moment and variance are also infinite.
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