, { x,1. {%) and {4) hat converge almost surely to a,b,c onsider three sequences of random variab...
Prove that if Xn is a sequence of random variables that converge to a limit X almost surely, then Xn converges to X in probability. Give an example to show that convergence in probability does not imply almost sure convergence. Suppose X bar is the mean of random sample of size 100 from a large population with mean 70 and standard deviation 20. Without use CLT , give the estimated probability P(65<x<75).
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 +...
2) Which of these sequences converge? 1 (ii) (ne "} SuAu C. (ii) and (ii) only B. (i) and (ii) only. A. All of them converge F. All of them diverge E. (ii) only D. (ii) only 2) Which of these sequences converge? 1 (ii) (ne "} SuAu C. (ii) and (ii) only B. (i) and (ii) only. A. All of them converge F. All of them diverge E. (ii) only D. (ii) only
Three random variables A, B, and C and 1. The random processes X(t) and Y (t) answer the questions below. (24 points) independent identically distributed (id) uniformly between are defined by the given equations. Use this information to are X(t) = At + B Y(t) = At + C (a) Find the autocorrelation function between X(t) and Y(t) (b) Find the autocovariance function between X (t) and Y(t). (c) Are X(t) and Y(t) correlated random processes? Three random variables A,...
8. Let {Xn, n = 1, 2, . . . } and (, , n = 1, 2, . . . } be two sequences of random variables, defined on the sample space Suppose that we know . Xn → X, G.8 Prove that XnYX+Y. 8. Let {Xn, n = 1, 2, . . . } and (, , n = 1, 2, . . . } be two sequences of random variables, defined on the sample space Suppose that...
number 4 Ex. 3.1. Determine, when possible, the function f(x) to which the following sequences converge in a pointwise way: sin(na) (ii) (i) n 2n (iii) 1 2n (iv) n (v) n! (vi) x
Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle 0 < x < a and 0 < y < b for a, b E R+. Show mathematically that X and Y are independent. Hint: (a) Recall JDx "lly f(r, y) dy dx-1 (b) Recall X, Y are independent if ffy fry Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle...
Suppose X1, X2,... are independent Geometric (number of trials) random variables where Xi ~ Geometric(p = 1/i^2) a) It is easily shown that Xn converges to a for some constant a. Name it. b) According to the Borel-Cantelli Lemmas, does Xn almost surely converge to a? Suppose Xi, X2, are independent Geometric (number of trials) random variables where x,~ Geometric(pal+) |. a) It is easily shown that Xa for some constant a. Name it. b) According to the Borel-Cantelli Lemmas,...
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...
onsider X is a continuous variable with the following density function f(x) = 2(x - 1) 1<x<2 otherwise (a) Compute expected value of the random variable X (b) Compute standard deviation of the random variable X (c) If g(x) = 4x + * 4X + 2, what is the expected value of g(x)? (d) What is the standard deviation of g(x)?