Problem 3 Suppose that Y. whose density is that Y Y sdom smumph from Y, is...

Question

Question

math Statistics-And-Probability

Answer 1

Answer #1

$E[Y^2] = \int_{0}^{1} y^2(1 - \theta (y - 1/2))~dy = \int_{0}^{1} y^2 - \theta (y^3 - y^2/2)~dy$

$= \left [ y^3/3 - \theta (y^4/4 - y^3/6) \right ]_{0}^{1} = 1/3 - \theta (1/4 - 1/6) = 1/3 - \theta /12$

Expectation of T is,

$E[T] = E\left [ 1 - (3/n) \sum_{i=1}^{n}Y_i^2 \right ] = 1 - 3 E\left [\sum_{i=1}^{n}Y_i^2/n \right ] = 1 - 3E[Y^2]$

$= 1 - 3\left ( 1/3 - \theta /12 \right ) = 1 - 1 + \theta /4 = \theta /4$

Answer 2

Similar Homework Help Questions

Suppose that X is a continuous random variable whose probability density function is given by (C(4x...

Suppose that X is a continuous random variable whose probability density function is given by (C(4x sa f(x) - 0 otherwise a) What is the value of C? b) Find PX> 1)
3. (a) Suppose that xi,... ,Vn are a random sample having probability density function Here α...

3. (a) Suppose that xi,... ,Vn are a random sample having probability density function Here α is restricted to be positive. Determine the MLE of a. (b) Suppose that r1,..., Jn are a random sample from a geometric distribution Here the parameter 0 < θ < 1. Determine the MLE of θ and show carefully that it is an MLE: it does not suffice to solve the score equation
Problem 1. Suppose that X and Y are jointly contimmous with joint probability density function re-r(1+y),...

Problem 1. Suppose that X and Y are jointly contimmous with joint probability density function re-r(1+y), İfx > 0 and y > 0. otherwise. (a) Find the marginal density functions of X and h (b) Calculate the expectation of E(XY). (e) Calculate the expectation E

Problem 4 Let Yı, Y2, ..., Y, denote a random sample from the probability density function...

Problem 4 Let Yı, Y2, ..., Y, denote a random sample from the probability density function (0 + 1)ye f(0) = 0 <y <1,0 > -1 elsewhere Find the MLE for .
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and...

3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.
6. Suppose that X and Y are jointly continuous random variables with joint density f(r, y)otherwise...

6. Suppose that X and Y are jointly continuous random variables with joint density f(r, y)otherwise (a) Given that X > 1, what is the expected value of Y? That is, calculate Ey X 〉 1).

(6) Suppose that X is an absolutely continuous random variable with density 1<I<2 f(3) = lo,...

(6) Suppose that X is an absolutely continuous random variable with density 1<I<2 f(3) = lo, otherwise. Find (a) the moment generating function MX(t). (b) the skewness of X (c) the kurtosis of X (7) Suppose that X, Y and Z are random variables such that p(X,Y) = 1 and p(Y,Z) = -1. What is p(X, Z)? Explain your answer. (8) Suppose that X, Y and Z are random variables such that p(X,Y) = -1 and p(Y,Z) = 0. What...
PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution -...

PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution - 5) +1 if 0 <r <1 fx(2) = 10 0. 0.w.. where @ € (-2, 2) is an unknown parameter. We define the estimate ēn as: ô, = 12X – 6 to estimate . (a) Is ên an unbiased estimator of e? (b) Is Ôn a consistent estimator of e?
1. Suppose X and Y are jointly continuous random variables with joint density function otherwise Let...

1. Suppose X and Y are jointly continuous random variables with joint density function otherwise Let U 2X-Y and V-2X +Y (i). What is the joint density function of U and V? (ii). Caleulate Var(UV)

Assumptions from problem #2 Problem 3: 10 points Continue with the same assumptions as in Problem...

Assumptions from problem #2 Problem 3: 10 points Continue with the same assumptions as in Problem 2. Recall that a random variable, Z, has the Gamma distribution with the density: fz (z) = λ2 z exp[-λ z] for z > 0, and fz(z) = 0, elsewhere. Conditionally given Z = z, a random variable, U, is uniformly distributed over the interval, (0, z) 1. Find conditional expectation. EZIU = ul. 2. Find conditional variance, VARZİU-ul 3. Find conditional expectation, E...