Let X1,., X, denote a random sample from the probability density function f(r) = 0x-1, 0 0. Show that X is a consistent estimator of
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- Suppose that Y1, . . . , Yn is a random sample from a population whose density function isSuppose the random variables X and Y have joint probability density function f(x,y) given by: (image)Find: P(X < Y) = fX|Y=y (x)Suppose that the random variables X and Y have a joint density function given by: f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise P(3 < X < 5, Y >1), P(X < 3), P(X +Y > 5), Find the joint distribution function (cdf),
- Verify that p(x) = 3x - 4 is a probability density function on [1, oo)and calculate its mean value.Let X1, X2, . . . , Xn be an i.i.d. random sample from a Beta distribution with density: f(x; θ) = Γ(2θ) Γ(θ) 2 x θ−1 (1 − x) θ−1 , 0 < x < 1, θ > 0. Find a sufficient statisticSuppose that the continuous random variable X has CDF Fx(X) = {(x-2)/x , x>2 and 0, x=<2} a. Determine, and sketch, the pdf (probability density function) of X. b. Find the mean and variance of X c. Determine the pdf of the random variable Y=X^2