代写DISTRIBUTIONS ON Rd代做留学生Matlab编程

DISTRIBUTIONS ON Rd

EXAMPLES FOR STUDY & PRACTICE

1. Continuous approximation of rectangles on (Rd, Bd) [Prop.4.3.1, p.35]

Given any > 0, any probability P on Bd, and any a < b in Rd , for each of the four types of rectangle: (a, b], [a, b),(a, b), [a, b] Rd, there is a δ > 0 with P ( (a±δ1, b±δ1]) = 0 for which we get all four approximations simultaneously

2. Suppose that (X, Y ) ~ unif(C) where C = (0, 1]2+(1, 2]2 in R2.

a) Determine the stochastic behaviour of each of the coordinate variables, X and Y .

b) Obtain both the conditional probability P(X ≤ 3/2 | Y ≤ 1/2) and corresponding marginal probability P(X ≤ 3/2).

c) Evaluate the correlation coefficient ρ(X, Y ).

3. statistical independence w. the uniform.

a) For U ~ unif(C), 0 < vol(C) < ∞ with U = (X,Y) on Rd = RrxRs, verify that, if C open & connected, then

b) U ~ unif(C) with U1,...,Ud (mutually) statistically independent then C open & connected C = (a, b) for some a < b in Rd.

4. uniform. in the unit ball:   X ~ unif(Bn), Bn = { t | |t| < 1 }

a) Determine the distribution function, FR(r), of the radial distance R = |X| and thus describe the distribution of Rn.

b) Uniformity can be deceptive in higher dimensions: if n is large, the random variable X will spend most of its time far from the origin, very close to its own boundary, Sn−1. Obtain asymptotic values of the mean and the standard deviation, ER & σ(R), as well as the first value of n such that P(R>.99) > .99.

c) Since we already know that X2 ~ Dn(1/21; 1), it is perfectly straight-forward to obtain the correlation thus to verify directly that X1,...,Xn are not statistically independent.

5. Suppose

where

T = Z+W, Z ~ G(1), W ~ G(2) and Z ?? W.

a) Obtain E|X|, EX and σ(X).

b) Let and determine EY and σ(Y).

c) Obtain the correlation ρ(X, Y).

6. Suppose that (X, Y) is uniformly distributed on the triangle with vertices (0, 0), (1, 1), (−1, 1)

a) Obtain EX, σ(X) and EY, σ(Y )

b) Determine the correlation ρ(X, Y).

c) Obtain the distribution of the radial distance

7. Lebesgue linear spaces in Rk: the multivariate Lp-spaces, p > 0

Establish the following basic facts

a) For every p > 0 and k 2 N, is a vector space, i.e. < Rk.

b) For any k ∈ N, if 0

c)

8. Lebesgue linear spaces in Rk continued: (cf.Chpt.6, p.27)

a) For any X ∈ , show that |EX| ≤ E|X| and indicate precisely the circumstances for equality. (recall Chpt.5, ESP4d))

b) For any X ∈ , show that |EX|2 ≤ E|X|2 and, just as above, indicate precisely the circumstances for equality.