Please answer all the parts neatly with all details.
Please answer all the parts neatly with all details. 6. Two independent random variables X and...
Please answer all the parts neatly with all details.
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi oo. Let Yn = (|X1| .+ |Xn|)/n. (a) Show that Yn ->v in probability. (b) Show that E(Y,) -- v. (c) Show that E(|X, - /u|) -0 where u = E(X)
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi...
Please answer all the parts neatly with all details.
4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
Please answer all the parts neatly with all details.
8. Let X1, X2, ... be an i.i.d. with Xn Let Y min(X,... , Xn) + 1 and Zn = max(X1,... , Xn) - 1. (a) Show that Y, - 0 and 0 - Z, have the same distribution uniform(0 1,0+ 1) (b) Show that Y, -P>0. (c) Show that n(Yn - Zn) converges in distribution and specify the limit distribution
L.11) Sums of independent random variables a) If X1 , X2 X, , , Xn are independent random variables all with Exponential μ distribution, then what is the distribution of XII + 2 +X3 + .tX b) If X is a random variable with Exponential[u] distribution, then what is the distribution of x +X1? c) If X1 , X2 , Х, , , X are independent random variables all with Normal 0. I distribution, then what is the distribution of...
2. Let X1 and X2 be independent Poisson random variables with parameters λ1 and A2. Show that for every n 21, the conditional distribution of X1, given Xi X2n, is binomial, and find the parameters of this binomial distribution
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X is m and Y is a zero-mean random variable. We define random variable V as V- VX2 +Y2. Show that: 0 <0 Where er cos "du is called the modified Bessel function of the first kind and zero order. The distribution of V is known as the Ricean distribution. Show that, in the special case of m 0, the Ricean distribution simplifies...
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
4a). Let X1 and X2 be independent random variables with a common cumulative distribution function (i.e., c.d.f.) F(y) = { 0" if0cyotherwise。 Find the p.d. f. of X(2,-max(X, , xa). Are X(1)/X(2) and X(2) independent, where X(1,-min(X,, X2) ?
number2 how to solve it?
Are x1 and x2 independent
- yes, they are independent.
Random variables X and Y having the joint density 1. 8 2)u(y 1)xy2 exp(4 2xy) fxy (x, y) ux- _ 3 1 1 Undergo a transformation T: 1 to generate new random variables Y -1. and Y2. Find the joint density of Y and Y2 X3)1/2 when X1 and X2 (XR 2. Determine the density of Y are joint Gaussian random variables with zero means...
8. We say that two discrete random variables X and Y , are independent when P(X = a, Y = b) = P(X = a)P(Y = b) for all a and b in the corresponding sample spaces. Let Xị and X, be independent Poisson random variables with parameters l1 = 3 and dy = 2 respectively. Find the probability of the event that X1 + X2 = 3. Hint: Since {X1 + X2 = 3} = {X} = 0, X2...