E303A] Digital System Design/(2018 Spring) Due time: 06/01 16:00 5. 8-segment de
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Question
E303A] Digital System Design/(2018 Spring) Due time: 06/01 16:00 5. 8-segment decoder for 8 symbols. Implement (draw logic diagram) the segment 4 of the 8-segment (a) Using K-map [3 pts] (b) Using a 3-8 decoder and OR gates. (5 pts) (c) Using 8-t0-1 multiplexer. [17pts decoder for 8 symbols Notes: output q. Each bit of q corresponds to one of the eight segments of a display according to the following pattern: 1.A eight-segment decoder is a combinational circuit with a three-bit input a and a 8-bit 2.The relationship between the 3-bit input a and one of 8 symbols is as follows: 111?? 80 a-000-?001 ?? 010-4011?100?101?110 ?Explanation / Answer
Solution Q1
Given X1, X2 is a random sample from the population of X ~ N(µ, ?2),
X1, X2 are iid with E(Xi) = µ and V(Xi) = ?2…………………………………….(1)
Back-up Theory
A statistic T is an unbiased estimator of a parameter µ, if E(T) = µ……………………….(2)
E(aX + bY + cZ) = aE(X) + bE(Y) + cE(Z) ……………………………………………….(3)
V(aX + bY + cZ) = a2V(X) + b2V(Y) + c2V(Z) ………………………………………….(4)
Part (a)
Given, µ1cap = (2/3)X1 + (1/3)X2, µ2cap = (1/4)X1 + (3/4)X2, and µ3cap = (1/2)X1 + (1/2)X2
E(µ1cap) = (2/3)E(X1) + (1/3)E(X2) [vide (3) under Back-up Theory]
= (2/3)µ + (1/3)µ [vide (1)]
= µ
=> µ1cap is an unbiased estimator of a parameter µ. ANSWER 1
[vide (2) under Back-up Theory]
E(µ2cap) = (1/4)E(X1) + (3/4)E(X2) [vide (3) under Back-up Theory]
= (1/4)µ + (3/4)µ [vide (1)]
= µ
=> µ2cap is an unbiased estimator of a parameter µ. ANSWER 2
[vide (2) under Back-up Theory]
E(µ3cap) = (1/2)E(X1) + (1/2)E(X2) [vide (3) under Back-up Theory]
= (1/2)µ + (1/2)µ [vide (1)]
= µ
=> µ3cap is an unbiased estimator of a parameter µ. ANSWER 3
[vide (2) under Back-up Theory]
Now, to determine the comparative efficiency of these 3 estimators,
V(µ1cap) = (2/3)2V(X1) + (1/3)2V(X2) [vide (4) under Back-up Theory]
= (4/9)?2 + (1/9)?2 [vide (1)]
= (4/9)?2 …………………………………………………………………………………(5)
V(µ2cap) = (1/4)2V(X1) + (3/4)2V(X2) [vide (4) under Back-up Theory]
= (1/16)?2 + (9/16)?2 [vide (1)]
= (5/8)?2 …………………………………………………………………………………(6)
V(µ3cap) = (1/2)2V(X1) + (1/2)2V(X2) [vide (4) under Back-up Theory]
= (1/4)?2 + (1/4)?2 [vide (1)]
= (1/2)?2 …………………………………………………………………………………(7)
(1), (2) and (3) => V(µ1cap) < V(µ3cap) < V(µ2cap)
=> µ1cap is the best estimator of a parameter µ. ANSWER 4
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