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Chapter 5: Problem 39

Let \(x\) be a Poisson random variable with mean \(\mu=2 .\) Calculate these probabilities: a. \(P(x=0)\) b. \(P(x=1)\) c. \(P(x>1)\) d. \(P(x=5)\)

Short Answer

Expert verified
Question: Calculate the probabilities for a Poisson random variable with a mean of 2 in the following scenarios: a. P(x=0) b. P(x=1) c. P(x>1) d. P(x=5) Answer: a. P(x=0) = e^(-2) b. P(x=1) = 2e^(-2) c. P(x>1) = 1 - (e^(-2) + 2e^(-2)) d. P(x=5) = (32e^(-2))/120

Step by step solution

01

Calculate \(P(x=0)\)

Using the Poisson probability formula and plugging in the given values, we get: $$ P(x=0) = \frac{e^{-2} 2^0}{0!} = e^{-2} $$ Evaluating this expression gives us the probability for \(x=0\).
02

Calculate \(P(x=1)\)

Using the Poisson probability formula again and plugging in the given values, we get: $$ P(x=1) = \frac{e^{-2} 2^1}{1!} = 2e^{-2} $$ Evaluating this expression gives us the probability for \(x=1\).
03

Calculate \(P(x>1)\)

Using the complement rule, we get: $$ P(x>1) = 1 - (P(x=0) + P(x=1)) = 1 - (e^{-2} + 2e^{-2}) $$ Evaluating this expression gives us the probability for \(x>1\).
04

Calculate \(P(x=5)\)

Again, using the Poisson probability formula and plugging in the given values, we get: $$ P(x=5) = \frac{e^{-2} 2^5}{5!} = \frac{32e^{-2}}{120} $$ Evaluating this expression gives us the probability for \(x=5\). Now we have calculated all the probabilities required for the exercise.

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