Find the Density FunctionDate: 01/24/2002 at 10:07:31 From: Tanya Subject: Probability Suppose X1, X2, X3 are independent and uniform on(0,1). find the density function of S = X1 + X2 + X3. Date: 01/24/2002 at 19:15:11 From: Doctor Anthony Subject: Re: Probability I shall use x, y, z rather than x1, x2, x3. You can obtain the distribution function of S by finding the volume of the unit cube cut off by the plane x + y + z = s The plane will be symmetrically positioned cutting the coordinate axes at (s,0,0), (0,s,0), (0,0,s) The volume of the tetrahedron cut off by the plane x + y + z = s is Vol = (1/3) area of base x perpendicular height (1/3)(1/2 x s^2) (s) = (1/6)s^3 So F(s) = s^3/6 for 0 < s < 1 For 1 < s < 2 the volume is calculated as above but with the 3 small tetrahedra outside the unit cube subtracted. The volume of each of these exterior tetrahedra is (1/6)(s-1)^3, so together they have a volume of (1/2)(s-1)^3, and the volume of the cube cut off by the plane is (1/6)s^3 - (1/2)(s-1)^3 and so F(s) = (1/6)s^3 - (1/2)(s-1)^3 for 1 < s < 2 For 2 < s < 3 we get the volume cut off as 1 - (1/6)(3-s)^3 and so F(s) = 1 - (1/6)(3-s)^3 for 2 < s < 3 To obtain the pdf we must differentiate F(s). So we have f(s) = (1/2)s^2 for 0 < s < 1 = (1/2)s^2 - (3/2)(s-1)^2 for 1 < s < 2 = (1/2)(3-s)^2 for 2 < s < 3 - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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