標題:
Cumulative Distribution Func
發問:
Let X,Y be uniformly distributed U[0,1] X and Y are independent. Let Z = max(X,Y) / min(X,Y) a. Find and sketch the CDF of Z b. Find and sketch the PDF of Z. i know that the range of Z is 1<= Z <= inf but how to sketch it? what should the x-axis be?
最佳解答:
a. the CDF is defined by F(z) = P(Z =< z) = 1 - P(Z>z) For z=>1, P(Z>z) = P(X>zY or Y>zX) = P(X>zY)+P(Y>zX) The last equality holds because the two events are mutually exclusive. P(X>zY)=∫(y=0 to 1) P(X>zy) f(y) dy =∫(y=0 to 1) P(X>zy) dy (since Y is U(0,1), f(y)=1 on [0,1]) = ∫(y=0 to 1/z) 1-zy dy + ∫(y=1/z to 1) 0 dy =1/z - 1/2z = 1/2z By symmetry, P(Y>zX)=1/2z. Hence P(Z>z) = 1/2z+1/2z = 1/z Therefore, F(z)=1-1/z for z=>1, and F(z)=0 for z<1. Using this formular, you can easily sketch the curve of CDF. b. the PDF is obtained by differentiating the CDF. So f(z) = 1/z^2 for z=>1 and f(z)=0 for z<1. Hence you can sketch the PDF according to this formula. PS. using the notations above, the horizontal axis is z and the vertical axis is F(z) or f(z).
其他解答:
Let X,Y be uniformly distributed U[0,1] X and Y are independent. Let Z = max(X,Y) / min(X,Y) Let U=max(X,Y) V=min(X,Y) Using the formila of Order Statistics, the joint distribution of U, V is g(u,v)=2 where 01 The CDF is ∫(t = 1 to s) 1/t^2 dt =-1/t |(1,s) =1-1/s
Cumulative Distribution Func
發問:
Let X,Y be uniformly distributed U[0,1] X and Y are independent. Let Z = max(X,Y) / min(X,Y) a. Find and sketch the CDF of Z b. Find and sketch the PDF of Z. i know that the range of Z is 1<= Z <= inf but how to sketch it? what should the x-axis be?
最佳解答:
a. the CDF is defined by F(z) = P(Z =< z) = 1 - P(Z>z) For z=>1, P(Z>z) = P(X>zY or Y>zX) = P(X>zY)+P(Y>zX) The last equality holds because the two events are mutually exclusive. P(X>zY)=∫(y=0 to 1) P(X>zy) f(y) dy =∫(y=0 to 1) P(X>zy) dy (since Y is U(0,1), f(y)=1 on [0,1]) = ∫(y=0 to 1/z) 1-zy dy + ∫(y=1/z to 1) 0 dy =1/z - 1/2z = 1/2z By symmetry, P(Y>zX)=1/2z. Hence P(Z>z) = 1/2z+1/2z = 1/z Therefore, F(z)=1-1/z for z=>1, and F(z)=0 for z<1. Using this formular, you can easily sketch the curve of CDF. b. the PDF is obtained by differentiating the CDF. So f(z) = 1/z^2 for z=>1 and f(z)=0 for z<1. Hence you can sketch the PDF according to this formula. PS. using the notations above, the horizontal axis is z and the vertical axis is F(z) or f(z).
其他解答:
Let X,Y be uniformly distributed U[0,1] X and Y are independent. Let Z = max(X,Y) / min(X,Y) Let U=max(X,Y) V=min(X,Y) Using the formila of Order Statistics, the joint distribution of U, V is g(u,v)=2 where 0
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