Let X1, ..., Xn be a random sample from U(0, 3). Recall that this means fXi...

Let X1, ..., Xn be a random sample from U(0, 3). Recall that this means fXi (xi) = 1 3 , 0 < xi < 3, i = 1, ..., n, and that all Xi are mutually independent. Let X(1) ≤ X(2) ≤ ... ≤ X(n) be the order statistics of the random sample. Denote Y1 = X(1).

• Derive FXi (xi).

• Find FY1 (y) = P(Y1 ≤ y). Hint: Use the complement rule of probability. That is, P(Y1 ≤ y) = 1 − P(Y1 > y).

• Show that fY1 (y) = n 3 ∗ (1 − y 3 ) n−1 , 0 < y < 3.

• Show that E(Y1) = 3 (n+1) .

• Compute V ar(Y1), utilizing fact that E(Y 2 1 ) = 18 (n+1)(n+2)

• Compute P(Y1 ≤ 1).

• Assume n = 3. Denote Y2 = X(2), the median. Describe what you would do to find fY2 (y).

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orchestra answered 1 year ago

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