Question

Consider a Poisson random variable X with parameter λ=3.

What is the probability that X is within two standard deviations of its mean?

Answer 1

To find the probability that a Poisson random variable X with parameter λ=3 is within two standard deviations of its mean, compute the mean and standard deviation, then calculate the lower and upper bounds. Finally, use the Poisson cumulative distribution function to find the probability between the lower and upper bounds.

Step-by-step explanation:

To find the probability that a Poisson random variable X with parameter λ=3 is within two standard deviations of its mean, we need to compute P(X > μ - 2σ and X < μ + 2σ). The mean of a Poisson distribution is equal to its parameter, so in this case, the mean is 3.

Step 1:

Compute the mean and standard deviation.

Mean (μ) = λ = 3

Standard Deviation (σ) = sqrt(λ) = sqrt(3)

Step 2:

Compute the lower bound:

Lower bound = μ - 2σ = 3 - 2(sqrt(3))

Step 3:

Compute the upper bound:

Upper bound = μ + 2σ = 3 + 2(sqrt(3))

Step 4:

Compute the probability:

P(X > lower bound and X < upper bound) = P(X > 3 - 2(sqrt(3)) and X < 3 + 2(sqrt(3)))

To calculate this probability, we can use the Poisson cumulative distribution function (CDF) and subtract the probability of X being less than or equal to the lower bound from the probability of X being less than or equal to the upper bound.