Let U = {1,2,3,4,5,6,7,8,9); X= {2,4,6,8}; Y={2,3,4,5,6}; Z={1,2,3,8,9} XnY X' Yn(XuZ) (XnY') u (Z'nY') X'nZ X’n(Y’uZ) (XnY) u (X'nZ)
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- 1. Find the natural cubic spline sN (x) passing through the 3 points (xj, yj) given by (0, 2), (2, 3), and (3, 1).Then evaluate sN (1).1- Let X,,X,,X; have the joint pdf given by 1 flrx2,%3) =4 (2m)32 B g 0w 132,03 —5(xj+aj+x @ 2vae 3) ,—0 < X3,X5,X3 < © Then ¥ = 33, X7 ~(a) express ux, u y, and uz as func-tions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate ux, u y, and uz at the given point (x, y, z). u = e^(qr) sin-1 p, p = sin x, q = z^2 ln y, r = 1/z; (x, y, z) = (pai/4, 1/2, -1/2)
- . Show that it is possible to solve u and v from:xyu + xy2uv2 = 2 and x2yvu4 + yu3v2 = 2,in terms of x and y uniquely near the point (x, y, u, v) = (1, 1, 1, 1); also find the first four partial derivatives at the point (1, 1).If Z1, Z2, Z3 are independent and identically distributed, such that Zi--Geom(0.4) for i=1,2,3. What is P(Z1+Z2+Z3=7)?f X1,X2,...,Xn constitute a random sample of size n from a geometric population, show that Y = X1 + X2 + ···+ Xn is a sufficient estimator of the parameter θ.