We know that P(B1 ∩ B2) = 0, P(B1) = 1/4, P(B2) = 3/4, P(A) = 1/8. What is the maximum possible value of the product P(A | B1) · P(A | B2)?
Let B1, B2, B3 be pairwise exclusive, P(B1) = 1/8, P(B2) = 1/4, P(B3) = 5/8, P(A | B1) = 1, P(A | B2) = 1/3, P(A | B3) = 1/3. Calculate P(B2 | A).
(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent; similarly, let A and B2 be independent. Show that in this case, A and B1∪B2 are independent if and only if A and B1∩B2 are independent. (c) Given P(A) = 0.42,P(B) = 0.25, and P(A∩B) = 0.17, find (i)P(A∪B) ; (ii)P(A∩Bc) ; (iii)P(Ac∩Bc) ; (iv)P(Ac|Bc).
3. Consider the game illustrated by the payoff matrix below: Jeffrey B1 B2 -4,- 4 1 ,-6 Curtis A2 -6,1 0,0 b. Suppose that the game is repeated 10 times and assume a discount factor 8 (where 0< 8 < 1). i. [2] What does each player choose to do in the 10th round of play? ii. [2] Can Curtis and Jeffrey credibly commit to playing (A2, B2) in any of the rounds that they play? c. Suppose now that...
When performing an F-test, if the null hypothesis is H. : B1 = B2 = 0 what is the alternative hypothesis? (B1 < 0 and B2 > 0) or (B1 > 0 and B2 < 0) B1 + 0 and/or B2 + 0 O B1 + 0 and B2 + 0 O (B1 + 0 and B2 = 0) or (B1 = 0 and B2 + 0)
How was the linear transformation of b1 and b2 were applied
(L(b1) , L(b2))?
NOTE: b1=(1,1)^T , b2=(-1,1)^T
Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R? into itself and defined by where (bi, b2] is the ordered basis defined in Example 3. Find the matrix A represent- ing L with respect to [bi, b2l Solution Thus, A0 2 onofosmation D defined by D(n n' maps P into P, Given the ordered
Linear Transformations EXAMPLE 4 Let...
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3 if the number of heads is less than 49.5 and B2/3 otherwise. Bound the error probability using the Chernoff bound derived in lecture video (in its final form, after simplifcation).
A coin is equally likely to be either B1/3 or B2/3. To figure out the bias, we toss the coin 99 times and declare B1/3...
{(0, b), (1, b2),, , . (k, bk)) İs a set of k points in R2. Show that in the horizontal line 5. Suppose P of best fit f(x) A, A is the average of the numbers b1,. b
{(0, b), (1, b2),, , . (k, bk)) İs a set of k points in R2. Show that in the horizontal line 5. Suppose P of best fit f(x) A, A is the average of the numbers b1,. b
0 1 Let S span 1 1 1 0 }, a basis for S. Show that| (a) Let B1 { 1 0 1 1 0 is also a basis for S 0 B2 { 1 (b) Write each vector in B2 (c) Use the previous part to write each vector in B2 with respect to Bi (how many components should each vB, vector have?) (d) Use the previous part to find a change of basis matrix B2 to B1. What...
What is the p.m.f. (Probability Mass Function) for A1, where A1 equals B1 minus B2 and we are given that Bi ∼ Ber(p).
Assume that the transition matrix from basis B = {b1, b2, b3} to basis C = {c1, c2, c3} is PC,B = 1/2*[ 0 -1 1 ; -1 1 1 ; 1 0 0 ]. (a) If u = b1 + b2 + 2b3, find [u]C. (b) Calculate PB,C. (c) Suppose that c1 = (1, 2, 3), c2 = (1, 2, 0), c3 = (1, 0, 0) and let S be the standard basis for R 3 . (i) Find...