Use mathematical induction to show that P(A1∩A2∩...∩An) = P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩ An-1) You can assume that you know...
Use mathematical induction to show that
P(A1∩A2∩...∩An) =
P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩
An-1)
You can assume that you know P(A|B) = P(A|B)P(B)
Homework Answers
TOPIC:Probability and the method of mathematical induction.
Add Answer to:
Use mathematical induction to show that
P(A1∩A2∩...∩An) =
P(A1)P(A2|A1)P(A3|A1∩A2)....P(An|A1∩A2∩...∩
An-1)
You can assume that you know...
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation a...
: Let a1, a2, a3, . . . be the sequence of integers defined by a1 = 1 and defined for n ≥ 2 by the recurrence relation an = 3an−1 + 1. Using the Principle of Mathematical Induction, prove for all integers n ≥ 1 that an = (3 n − 1) /2 .
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Usi...
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and clearly describing the events At, A2 and A3. construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 01,02 and 03, say.
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and...
1. Suppose you are given 3 algorithms A1, A2 and A3 solving the same problem. You...
1. Suppose you are given 3 algorithms A1, A2 and A3 solving the same problem. You know that in the worst case the running times are Ti(n) = 101#nº + n, Ta(n) = 10”, TS(n) = 101 nº logo n10 (a) Which algorithm is the fastest for very large inputs? Which algorithm is the slowest for very large inputs? (Justify your answer.) (b) For which problem sizes is Al the best algorithm to use (out of the three)? Answer the...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may eq...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
A3 (use Find span of A1, A2, elementary row operations) 1 1 4. = [ 13.43=[1]...
A3 (use Find span of A1, A2, elementary row operations) 1 1 4. = [ 13.43=[1] A =6
Assume that in A1, A2, A3, and A4 you have the values of 1, 2, 3, and 4, respectively. In B1, C1, and D1, have the lette...
Assume that in A1, A2, A3, and A4 you have the values of 1, 2, 3, and 4, respectively. In B1, C1, and D1, have the letters a, b, and c, respectively. In B2, C2, and D2 you have the letters of d, e, and f, respectively. In B3, C3, and D3 you have the letters of g, h, and i, respectively. What will the command of =VLOOKUP(3,A1:D4,3) return? Group of answer choices h c a g i
LIDO D EDIUL 9. Let A1, A2, ... be a sequence of events. Show that PA...
LIDO D EDIUL 9. Let A1, A2, ... be a sequence of events. Show that PA A - A) = P(A) - PA A - UAA-2 UAA) for i = 2, 3,.... Hint: You don't need induction to prove this. You can assume, without proof, that A A-2 UA A-2 UAA = A (A-2 UA-2 UA) and A = AB Ü AB
(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;...
(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).
The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.45....
The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.45. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits. (a) Are A1 and A2 mutually exclusive? - Select your answer -YesNoItem 1 Explain your answer. The input in the box below will not be graded, but may be reviewed and considered by...
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and clearly describing the events At, A2 and A3. construct a 100(1-a)% joint confidence intervals for estima- tion of three parameters, denoted by 01,02 and 03, say.
10] Q3. (a) Prove the Bonferroni Inequality on three events A1, A2 and A3: P(An Agn A)21-P(A)- P(A2) - P(Aa) (b) Using the results in Q3.(a), and...
1. Suppose you are given 3 algorithms A1, A2 and A3 solving the same problem. You know that in the worst case the running times are Ti(n) = 101#nº + n, Ta(n) = 10”, TS(n) = 101 nº logo n10 (a) Which algorithm is the fastest for very large inputs? Which algorithm is the slowest for very large inputs? (Justify your answer.) (b) For which problem sizes is Al the best algorithm to use (out of the three)? Answer the...
(1) Let a (.. ,a-2, a-1,ao, a1, a2,...) be a sequence of real numbers so that f(n) an. (We may equivalently write a = (abez) Consider the homogeneous linear recurrence p(A)/(n) = (A2-A-1)/(n) = 0. (a) Show ak-2-ak-ak-1 for all k z. (b) When we let ao 0 and a 1 we arrive at our usual Fibonacci numbers, f However, given the result from (a) we many consider f-k where k0. Using the Principle of Strong Mathematical Induction slow j-,-(-1...
A3 (use Find span of A1, A2, elementary row operations) 1 1 4. = [ 13.43=[1] A =6
LIDO D EDIUL 9. Let A1, A2, ... be a sequence of events. Show that PA A - A) = P(A) - PA A - UAA-2 UAA) for i = 2, 3,.... Hint: You don't need induction to prove this. You can assume, without proof, that A A-2 UA A-2 UAA = A (A-2 UA-2 UA) and A = AB Ü AB
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