Let f(y, A) be the density function of a continuous random variable, Y, where A is the parameter of the distribution. (a) with reference to family of densities f(y, A) , define what a regular parametric family is. (b) Give three examples of regular parametric families.
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- b) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².Find a normal vector to the level curve f(x, y) = c at P. f(x, y) x² + y² 1 C = 2' P(1, 1)b) Let Z₁ = X-XN (0,1), and W₁ dx YHY~N(0,1), for i = 1,2,3,...,10, then: dy i) State, with parameter(s), the probability distribution of the statistic, T = - 54 1² ii) Find the mean and variance of the statistic T = Σ},wp? Σ1,2,3 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ẞ such that P(T> B) = 0.01, where T = Σ₁Z₁² + ₁ W₁².
- Find a normal vector to the level curve f (x, y) = c at P. f(x, y) = x/(x2 + y2) c = 1/2, P(1, 1)TC Let f(x) =- 4x - cos-1(x), then the equation | of the normal line to the curve at the point (0, 0) is A) y = -x. B) y =x. 3 C) y = 3x. D) у 3D — х+3 E) y = x + 3.Let X1, X2, . . . , Xn be iid from distribution: f(x; θ) = 2x/θ^2 , 0 < x < θ (a) Find a sufficient statistic for θ. (b) Is it minimal?
- i) Let (X1,...,Xn) be a random sample from P in a parametric family. Obtain moment estimators of parameters in the following cases. (i) P is the gamma distribution with shape parameter α>0 and scale parameter γ>0. (ii) P has Lebesgue density θ−1e−(x−a)/θI(a,∞)(x), a ∈R, θ>0. (iii) P has Lebesgue density Γ(α+β) Γ(α)Γ(β)xα−1(1−x)β−1I(0,1)(x), α>0, β>0. (iv) P is the log-normal distribution with parameter (µ,σ2) (i.e., logXLet (X,Y)' have density for r, y > 0, f (x, y) = (1+x)² (1+xy) > 0, otherwise. Show that X and X Y are independent, equidistriduted random variables and determine their distribution.Question 2 of 5 > Compute s F · dS for F = (y,5, x), S being the portion of the plane x + y + z = 1 in the octant x, y, z > 0, with an upward-pointing normal. (Use symbolic notation and fractions where needed.) F· dS =
- Let X and Y be continuous random variables having a joint pdf given by f(x, y) = e-*, 0sysx 3).Q1) Discrete joint variables X and Y with probability density f(x,y) (pdf) are given in this table. Find: 1) The Covariance Cov(X,Y)? (Cov(X,Y) = MxY-MxMY) 2) The correlation (pxy) between X and Y where Pxy = Cov(X,Y) PXPY Y 3 fx(x) f(x,y) 1 2 1 1/4 1/4 0 X 2 0 1/4 1/4 fy(y) Note that 2 n=2 n=3 Px² = Σn²±²x² f(x, y) - μ and py² = Σ3y² f(x, y) – µ Zk=01) The joint probability density function of the X and y random variables, fx,y (xiy)= cx,0SEE MORE QUESTIONS