Problem 5 Let N₁ (t) and N₂ (t) be two independent Poisson processes with rate ₁ and >₂ respectively. Let N(t) = N₁ (t) + N₂(t) be the merged process. Show that given N(t) N₁ (t) ~ Binomial (n,₁). = Note: We can interpret this result as follows: Any arrival in the merged process belongs to N₁(t) with probability and belongs to N₂(t) with probability 12 independent of other arrivals. X₁ +d₂ X₁ + λ2

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Problem 5
Let N₁ (t) and N₂ (t) be two independent
Poisson processes with rate ₁ and ₂
respectively. Let N(t) = N₁ (t) + N₂ (t) be the
merged process. Show that given N(t) = n,
N₁ (t) ~ Binomial (n,1₂).
Note: We can interpret this result as
follows: Any arrival in the merged process
and
belongs to N₁ (t) with probability
belongs to N₂(t) with probability
independent of other arrivals.
X₁
X₁ + A₂
12
X₁ + X2
Transcribed Image Text:Problem 5 Let N₁ (t) and N₂ (t) be two independent Poisson processes with rate ₁ and ₂ respectively. Let N(t) = N₁ (t) + N₂ (t) be the merged process. Show that given N(t) = n, N₁ (t) ~ Binomial (n,1₂). Note: We can interpret this result as follows: Any arrival in the merged process and belongs to N₁ (t) with probability belongs to N₂(t) with probability independent of other arrivals. X₁ X₁ + A₂ 12 X₁ + X2
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