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8. Problem 8. Let X be a Poisson random variable with parameter \lambda and Y an independent Bernoulli random variable with parameter p. Find the probability mass function of X + Y. Final Answers: (a) P(X + Y = z) = \begin{cases} (1 - p)e^{-\lambda} & z = 0\\ (1 - p)e^{-\lambda}\frac{\lambda^z}{z!} + pe^{-\lambda}\frac{\lambda^{z-1}}{(z - 1)!} & z \ge 1\\ 0 & \text{otherwise} \end{cases}

          8. Problem 8.
Let X be a Poisson random variable with parameter \lambda and Y an independent Bernoulli
random variable with parameter p. Find the probability mass function of X + Y.
Final Answers:
(a) P(X + Y = z) = \begin{cases} (1 - p)e^{-\lambda} & z = 0\\ (1 - p)e^{-\lambda}\frac{\lambda^z}{z!} + pe^{-\lambda}\frac{\lambda^{z-1}}{(z - 1)!} & z \ge 1\\ 0 & \text{otherwise} \end{cases}

8. Problem 8.
Let X be a Poisson random variable with parameter λand Y an independent Bernoulli
random variable with parameter p. Find the probability mass function of X + Y.
Final Answers:
(a) P(X + Y = z) = (1 - p)e^-λ z = 0
(1 - p)e^-λ(λ^z)/(z!) + pe^-λ(λ^z-1)/((z - 1)!) z ≥1
0 otherwise

Added by Kevin B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Jason Horton

Step 1

We can do this by considering all possible ways that X and Y can combine to give us a sum of z. If Y=0, then X must be equal to z. The probability of this happening is given by the Poisson probability mass function: P(X=z) = (e^-A * A^z) / z! If Y=1, then X Show more…

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Problem 8. Let X be a Poisson random variable with parameter Î» and Y an independent Bernoulli random variable with parameter p. Find the probability mass function of X + Y. Final Answers: (a) P(X+Y=z) = Î»^z * e^(-Î») / z! if z â‰¥ 1 1 - Î» * e^(-Î») if z = 0