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- Help me please: Let X be a random variable that is uniformly distributed over the interval [0, 2π] and let Y = cos(X);Z = sin(X). Show that Y and Z are not independent.The average diameter of ball bearings of a certain type is supposed to be 0.5 inch. What conclusion is appropriate when testing H0: μ = 0.5versus Ha: μ ≠ 0.5 inchin each of the following situations? (a) n = 11,t = 1.8,α = 0.05reject H0 fail to reject H0 (b) n = 11,t = −1.8,α = 0.05reject H0 fail to reject H0 (c) n = 26,t = −2.7,α = 0.01reject H0 fail to reject H0 (d) n = 26,t = −3.6reject H0 at any reasonable significance level fail to reject H0 at any reasonable significance levelp = m(1-dt) for d (buisness)
- b. At α = 0.05, test your hypothesis.Assuming X and Y have joint density f(x; y) = 1/4 for all 0<=x<=2 and 0<=y<=2, and f(x, y) = 0 otherwise.(i) Show that X and Y are independent random variables, and each has the uniform distribution on [0; 2]?(ii) what is E(X) and Var(X)?(iii) Compute Var(X + Y ) and Cov(X + Y, Y )?Given the random variables X and Y having the following joint density: f(x, y) = 2(x + y) for 0 < y < x < 1. A) Compute the conditional pdfs: fX│Y (x) and fY│X (y). B) Are X and Y independent?
- Consider X and Y are joint distributed with PDFf(x,y)=x+y, 0≤x≤1, 0≤y≤1. (a) Find the probability that X > 0.5. (b)FindP(X> Y 1/2).(c) Are X and Y independent?Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)An random experiment consists of marking a point randomly inside a half -circle with a radius of 1,as shown below. Let’s call this point N. Let X and Y be the random variables that denote thecoordinates such that X is the horizontal coordinate and Y is the vertical coordinate. Answer thefollowing question. Let g(X, Y ) be the area inside a circle that has the point N as its center and touchesthe half-circle at a single point, as shown below, in gray. Please note that the small circle canextend beyond the half circle on the bottom. Write g(X, Y ) and specify its range. Pleaseprovide the solution step by step.
- The probability that Paul’s train to work is late on any day is 0.15, independently of other days. (i) The number of days on which Paul’s on which Paul’s train to work is late during a 450-days period is denoted by the random carriable Y. Find a value of a such the P(Y>a)≈ 1/6 . In the expansion of (0.15+0.85)^50 , the terms involving 0.15^r and 0.15^(r+1) and denoted by Tr and Tr+1 respectively. (ii) Show that Tr / Tr+1 = 17(r+1)/3(50-r)The joint pdf of X and Y is given by fx,y(x,y) = {4x2 0<y<x<1} {0 otherwise} Let U =g(X,Y) = Y-X2. What is support of U?