Example: Let us consider the geometric distribution with parameter p € (0, 1). Then N = N, Ƒ = 2º and P(A) = ΣkɛAP(1 − p)k-1. The power set ♬ is clearly a sigma-algebra since it contains every subset of N. We will show that P satisfies Kolmogorov's axioms. First of all P is a map from ♬ to [0, 1]. Secondly ∞ k-1 P(N) = Σp(1 − p)k−¹ = pΣ(1 − p)² = k=1 k=0 An Σp(1-p)k-1 kЄUn=1 An 1 k-1 Aį disjoint - and third let us consider a sequence of disjoint events A₁, A2,.... Then P (Ũ.A.) = \n=1 P - p) = 1 ∞ Σ Σ p(1 − p)k-¹ = ΣP(An). n=1 k€ An n=1
Example: Let us consider the geometric distribution with parameter p € (0, 1). Then N = N, Ƒ = 2º and P(A) = ΣkɛAP(1 − p)k-1. The power set ♬ is clearly a sigma-algebra since it contains every subset of N. We will show that P satisfies Kolmogorov's axioms. First of all P is a map from ♬ to [0, 1]. Secondly ∞ k-1 P(N) = Σp(1 − p)k−¹ = pΣ(1 − p)² = k=1 k=0 An Σp(1-p)k-1 kЄUn=1 An 1 k-1 Aį disjoint - and third let us consider a sequence of disjoint events A₁, A2,.... Then P (Ũ.A.) = \n=1 P - p) = 1 ∞ Σ Σ p(1 − p)k-¹ = ΣP(An). n=1 k€ An n=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
please if able explain the following proof in more detail especially the yellow parts, where does p(1-p)k-1 come from?
![Example: Let us consider the geometric distribution with parameter p € (0,1). Then N = N,
F = 22 and P(A) = ΣkɛAP(1 − p)k-1. The power set F is clearly a sigma-algebra since it
contains every subset of N. We will show that P satisfies Kolmogorov's axioms. First of all P
is a map from F to [0, 1]. Secondly
k-1
P(N) = Σp(1 − p)k−¹ = pΣ(1 − p)k =
1
k=1
k=0
Р
p)
and third let us consider a sequence of disjoint events A₁, A2, .... Then
1
∞
P An
(ŨA.) = Σ P(1-p)k-1 Av dijoint ¶ Σ P(1 − p)²-¹ = [P(A₂)|
k-1
Σ − ΣP(An).
\n=1
ke Un=1 An
n=1 k€ An
n=1
The natural question is whether in general we can always assign a probability for every possible
subset ACN? We will see in the next subsection that this is not possible for general sample
spaces Ω.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e237d3f-b8e6-4775-a6f9-5671b153aef2%2Fd1a7165e-b077-46db-ace4-26ef69eabae1%2F8kfah7_processed.png&w=3840&q=75)
Transcribed Image Text:Example: Let us consider the geometric distribution with parameter p € (0,1). Then N = N,
F = 22 and P(A) = ΣkɛAP(1 − p)k-1. The power set F is clearly a sigma-algebra since it
contains every subset of N. We will show that P satisfies Kolmogorov's axioms. First of all P
is a map from F to [0, 1]. Secondly
k-1
P(N) = Σp(1 − p)k−¹ = pΣ(1 − p)k =
1
k=1
k=0
Р
p)
and third let us consider a sequence of disjoint events A₁, A2, .... Then
1
∞
P An
(ŨA.) = Σ P(1-p)k-1 Av dijoint ¶ Σ P(1 − p)²-¹ = [P(A₂)|
k-1
Σ − ΣP(An).
\n=1
ke Un=1 An
n=1 k€ An
n=1
The natural question is whether in general we can always assign a probability for every possible
subset ACN? We will see in the next subsection that this is not possible for general sample
spaces Ω.
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