7.6. Let A1, A2, ..., An be arbitrary events, and define Ck = {at least k of the Ai occur}. Show that n n Σ P(Gk) = Σ PCAK) k = 1 k = 1
7.6. Let A1, A2, ..., An be arbitrary events, and define Ck = {at least k of the Ai occur}. Show that n n Σ P(Gk) = Σ PCAK) k = 1 k = 1
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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why they area equal to E[X]? could you explain step by step more specifically and detailedly, such as using the definition of E[X]?
![7.6. Let A₁, A₂, ..., An be arbitrary events, and define
Ck = {at least k of the A; occur}. Show that
n
n
Σ P(CK) = Σ P(AK)
k = 1
k = 1
Hint: Let X denote the number of the A, that occur. Show that both sides of
the preceding equation are equal to E[X].
let X denote the mumber of Az that occur
I PLAK) = P(A₁) +P(A₂) +
P(An)
kal
= E [ number of Ar occurs ]
-E[x]
и
Σ P(CK) =
k=1
Σ PL at least k of the Ai occer]
ㅑ기
E I member of Az occurs ]
= {[x]
и
Thus, & P(AK)= £ P (4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F55ef11ec-a56d-4e20-a638-de429c981cb2%2F85b0a63f-2a3f-4b07-89a4-8545a4631eca%2Fu2h1zzv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7.6. Let A₁, A₂, ..., An be arbitrary events, and define
Ck = {at least k of the A; occur}. Show that
n
n
Σ P(CK) = Σ P(AK)
k = 1
k = 1
Hint: Let X denote the number of the A, that occur. Show that both sides of
the preceding equation are equal to E[X].
let X denote the mumber of Az that occur
I PLAK) = P(A₁) +P(A₂) +
P(An)
kal
= E [ number of Ar occurs ]
-E[x]
и
Σ P(CK) =
k=1
Σ PL at least k of the Ai occer]
ㅑ기
E I member of Az occurs ]
= {[x]
и
Thus, & P(AK)= £ P (4)
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