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- 1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.1. Let the combined density function of the two -dimensional continuous random variable (x, y) are p (x, y) = {1, | y | <x, 0 <x 10, other condition density functions p (x | y) And condition expectation E (x | y = y)
- A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0Let X and Y be two continuous random variables with joint probability density function f(x,y) = 2xy for 0 < x < y < 1. Find the covariance between X and Y.If the joint probability density function of two continuous random variables X and Y isgiven byf(x; y) = 2, 0 < y < 3x, 0 < x < 1; find(a) f(yjx),(b) E(Y jx),(c) Var(Y jx).
- b) Let Y1, Y2, ..., Yn be a random sample from a population with probability density function in part a). Show that the best test for the hypothesis in part a) rejects Ho if |yi <c i=1 where c solves the probability equation a = P(II1Yis c[0 = 2). c) Let X1,X2, ..., Xn be a random sample from GAMMA(2,ß) distribution, and consider Y = E-,Xi- Show whether or not Y is a pivotal quantity and give its distribution.5. Two random variables X and Y have a joint probability density function 0 < y < x < 2 fxy(x, y) = elsewhere (a) Find the marginal density functions of X and Y. (b) Are X and Y statistically independent?5. Suppose Z is a Gaussian random variable with mean 0 and 1. (a) Find the moment generating function of Z2. (b) Find the density function of Z2.
- 7)Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: X~ f (x I θ) . Find the distribution function of the random variable X.Verify that p(x) = 3x - 4 is a probability density function on [1, oo)and calculate its mean value.A tour is scheduled to start at 5:00 p.m., 5:30 p.m., and 6:00 p.m. Once the tour starts, the gate closes. A visitor arrives at time x with probability density function f(x) = 1/(b-a) where a is 4:30 p.m. and b is 6:00 p.m. Determine the probability that a visitor will wait less than 10 minutes for a tour.
