Problem # 1: Prove that Pr(AB)-1-Pr(A, 1B). Problem#2: Show that if E and F are independent,...
Suppose E and F are independent events. Find Pr[E′∩F] if Pr[E]=1/3 and Pr[F]=1/3 A and B are independent events. If Pr(A∩B)=0.24 and Pr[A]=0.3, what is Pr[B]?
Problem 1-4. Let AABC be a right triangle with hypotenuse AB. Suppose that D, E, F e AB, BF ZACB. Prove that ZDCE ZECF. FA, ZBDC is a right angle, and CE is the bisector of В E 14 F onu Rnonocitions 1.31 on these-but be sure to label what IT.
answer C1 and C2 then Prove Proposition 3.11 (Segment Subtraction): If A * B * C, D * E * F, AB s. DE, and em C2. Prove Proposition 3.12: Given AC DE. Then for any point B between A and C there is Group C (choose two) Problem Ci Propositi a unique point E between D and F such that AB Problem C3. Prove the first case of Propositi exists a line through P perpendicular to e. DE. on...
e and E and P events associated with S. Suppose that Pr(E)-0.5, Pr(F) -0.4 (a) If E and F are independent, calculate: i. Pr(EnF) ii. Pr(EUF) iii. Pr(El) iv. Pr(FIE) (b) If E and F are mutually exclusive, calculate: i. Pr(ENF) ii. Pr(EUF) iii. Pr(E|F) iv. Pr(FIE)
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
b. If E is the midpoint of CD and F is the midpoint of AB, show that EF is perpendicular to both AB and CD. b. If E is the midpoint of CD and F is the midpoint of AB, show that EF is perpendicular to both AB and CD.
E, F, and G in a sample space S. Assume that Pr[E]=0.5, Pr[F]=0.45, Pr[G]=0.55, Pr[E∩F]=0.3, Pr[E∩G]=0.3,and Pr[F∩G]=0.25. Find the following probabilities Pr[E∪F] = Pr[F′∩G]= Pr[E′∩G′]=
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
<C. Problem 1. For all x E R prove that r = 0 if V(e> 0) : Problem 2. For each of the below properties, name a function f: IRR that does not satisfy the property and prove your answer. (d) 3(e>0) 0) : Problem 2. For each of the below properties, name a function f: IRR that does not satisfy the property and prove your answer. (d) 3(e>0)
Show that if //AB-1//=E<1, then//A^-1-B//<=//B//(E/(1-E)) 16. Show that if | AB-1|| = < < 1, then 14--BISHBI