4. Let X1, X, be two r.v.'s with m.g.f. given by M(t1, tz)=[] (en+2 +1)+] (e? +e?)]”, t1, tz € R. Calculate E(X1), oʻ(X,) and C(X1, X2), provided they are finite.
CALCULATE C(X1, X2)
4. Let X,,X, be two r.v.'s with m.g.f. given by M(t,,ty)=[} (+2 +1) ++ (e? +e?)]”, tų, tz € R.
"k)-T, E(X"k+1)-0, k = 0.1, m.g.f. of X and also its ch.f. Then deduce the distribution of X. 6. Let X be a r.v. such that E(X Find the
Let X be a r.v. with probability density function f(x)-e(4-x2), -2 < otherwise (a) What is the value of c? (b) What is the cumulative distribution function of X? (c) What is EX) and VarX
pd.f. f given byf(x) = λe-d-a), X 1. Let Y be a rv. with M(t) for those t's for which it exists and also its chf. φ. Then calculate E(X), σ-(X), provided they are finite. α. Find its mg.f.
Let X1 and X2 be two discrete random variables, where X1 can
attain values 1, 2, and 3, and X2 can attain values 2, 3 and 4. The
joint probability mass function of these two random variables are
given in the table below: X2 X1 2 3 4 1 0.05 0.04 0.06 2 0.1 0.15
0.2 3 0.2 0.1 0.1 a. Find the marginal probability mass functions
fX1 (s) and fX2 (t). b. What is the expected values of X1...
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση! (Xi-X)2 and S2-n-T Ση#1 (Xi-X)2 be the estimators of σ2 (i) Show that the MSE of s is smaller than the MSE of S2 (ii) Find E [VS2] and suggest an unbiased estimator of σ.
We roll two dice. Assume all 36 possibilities are equally likely. Let X1 and X2 be the result of the first and second die, respectively. Let S be the sum of the scores, that is S = X1 + X2. Calculate the following: (a) P(S = k), for k 2,3,... 12. (b) P(X1 = 2 S = k) for k = 2,3,... 12. (c) P(X1 = 6|S = k) for k = 2, 3, ... 12.
O. Let X1 and X2 be two random variables, and let Y = (X1 +
X2)2. Suppose that E[Y ] = 25 and that the variance of X1 and X2
are 9 and 16, respectively.
O. Let Xi and X2 be two random variables, and let Y = (X1 X2)2. Suppose that and that the variance of X1 and X2 are 9 and 16, respectively E[Y] = 25 (63) Suppose that both X\ and X2 have mean zero. Then the...
(3) Let X = (X1, X2) be a two-dimensional random vector with variance Var[X= 121 12] Compute Covſa, Xi +a X2, 6, X1 + b2 X2], where an, az, bi, by are given constants.