Question
Let E1, E2, E3 and E4 be arbitrary events. Write the following events in set notations: i) not more than one of E1, E2, E3, E4.
ii) one and only one of E1, E2, E3, E4.
iii) E1 and at least one of E2, E3, E4.
iv) none of E2, E3 and E4 using E1.
b) Let the probability density function of r.v. X be
and if
and
, find Cov(u, v). Also check the independence of u and v.
Answer :
Word Count : 476
i) Not more than one of (E_1, E_2, E_3, E_4) means either zero or one of the events occurs. In set notation: [ (E_1^c \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2 \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3 \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3^c \cap E_4) ] ii) One and only one of (E_1, E_2, E_3, E_4) is exactly the same as the part of (i) excluding the zero-event case: [ (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; _________ _______ __________ __________ ___ ____ ____ ___ _________ _____ ____.
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i) Not more than one of (E_1, E_2, E_3, E_4) means either zero or one of the events occurs. In set notation: [ (E_1^c \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2 \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3 \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3^c \cap E_4) ] ii) One and only one of (E_1, E_2, E_3, E_4) is exactly the same as the part of (i) excluding the zero-event case: [ (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; _________ _______ __________ __________ ___ ____ ____ ___ _________ _____ ____.
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