a) here Au B ={x|2<=x<7} (as it contains all element which are in A or B or both)
b) A n B ={x|3<x<5}
c) A' n B' ={x|0<x<2 or 7<x<12}
d)A' u B'={x|0<x<3 or 5<x<12}
3. Suppose that A and B are two events defined over the same sample space, with probabilities P(A) 3/4 and P(B)- 3/8. (a) Show that P(A UB) 2 3/4. (b) Show that 1/8 < P(AB) 3/8 (c) Give inequalities analogous to (a) and (b) for P(A) 2/3 and P(B)1/2.
#5 (4 pts.) Consider the following sample space S and events A and B. s-(-4 < x < 2, 6 < x < 12), A={-4 < x < 0}, B=(-1 x<2), A and B are: a. (mutually exclusive, independent) b. (mutually exclusive, dependent) c. (non-mutually exclusive, independent) d. (non-mutually exclusive, dependent) #6 (4 pts.) In problem #5 P(B-A)- c. 1/4 d. 1/6
1.14 Consider events ArAg, Avon a sample space Ω. (a) Suppose A, c A-... c AN . Evaluate P(AIA)for i < j and for i > (b) Evaluate the set CAnd D1 A (c) Prove/Disprove: N-1 n AN ) = 1.
4. Let X1,..., X, be a random sample from a population with pdf 0 otherwise Let Xo) <...Xn)be the order statistics. Show that Xu/Xu) and X(n) are independent random variables
Consider the following pdf: ; 0<x<1 f(x)-2k ; l<x<2 0 otherwise (i)Determine the value of k. (ii) Find P(X 0.3) (iii) Find (0.1 〈 X 1.5).
1. Consider two independent events, A and B, where 0< P(A) <1,0< P(B)< 1. Prove that A and B' are independent as well.
Consider two independent events, A and B, where 0くP(A) < 1,0くP(8)く1. Prove that A' and B' are independent as well.
5) In free space, D 2ya,+4xya, - az mC/m2. Find the total charge stored in the region 1 <x < 2,1<y< 2, -1<z<4.
Let pdf of a r.v. X be given by f(x) = 1, 0<x< 1. Find Elet).
Consider fx (x)=e*, 0<x and joint probability density function fx (x, y) = e) for 0<x<y. Determine the following: (a) Conditional probability distribution of Y given X =1. (b) ECY X = 1) = (c) P(Y <2 X = 1) = (d) Conditional probability distribution of X given Y = 4.