Chapter 6: Q.6.17 (page 276)

Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X₁ + X₂ and Y = X₂ + X₃. The random vector X, Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n, Y = m}.

Short Answer

Expert verified

$P (X = n, Y = m) = e^{- (λ_{1} + λ_{2} + λ_{3})} \sum_{k - 0}^{m i n (n, m)} \frac{{λ_{1}}^{n - k}}{(n - k)!} . \frac{{λ^{k}}_{2}}{k!} . \frac{{λ_{3}}^{m - k}}{(m - k)!}$

Step by step solution

Content Introduction

Since X and Y are sums of independent Poisson variables, we have that $X ~ P o i s (λ_{1} + λ_{2}), Y ~ P o i s (λ_{2} + λ_{3})$

Content Explanation

Use conditional probability to obtain that

$P (X = n, Y = m) = \sum_{k = 0}^{\infty} P (X = n, Y = m, X_{2} = k) P (X_{2} = k) = \sum_{k = 0}^{\infty} P (X_{1} + X_{2} = n, X_{2} + X_{3} = m / X_{2} = k) P (X_{2} = k) = \sum_{k = 0}^{n} P (X_{1} = n - k) P (X_{3} = m - k) P (X_{2} = k) = \sum_{k = 0}^{n} \frac{{λ_{1}}^{n - k}}{(n - k)!} e^{- λ} . \frac{{λ_{3}}^{m - k}}{(m - k)!} e^{- λ_{3}} . \frac{{λ_{2}}^{k}}{k!} e^{- λ_{2}} = e^{- (λ_{1} + λ_{2} + λ_{3})} \sum_{k = 0}^{n} \frac{{λ_{1}}^{n - k}}{(n - k)!} . \frac{{λ_{3}}^{m - k}}{(m - k)!} . \frac{{λ_{2}}^{k}}{k!}$

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Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X₁ + X₂ and Y = X₂ + X₃. The random vector X, Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n, Y = m}.

Short Answer

Step by step solution

Content Introduction

Content Explanation

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Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X1 + X2 and Y = X2 + X3. The random vector X, Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n, Y = m}.

Short Answer

Step by step solution

Content Introduction

Content Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Suppose that Xi, i = 1, 2, 3 are independent Poisson random variables with respective means λi, i = 1, 2, 3. Let X = X₁ + X₂ and Y = X₂ + X₃. The random vector X, Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n, Y = m}.