Question

Suppose a random variable, x, follows a Poisson distribution. Let μ = 2.5 every minute, find the P(X ≥ 125) over an hour.

Answer 1

Using the normal approximation for the Poisson distribution, it is found that:

P(X ≥ 125) = 0.9812.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• The Poisson distribution can be approximated to the normal distribution with standard deviation as the square root of the mean.

For this problem, the measures are given by minute, hence the mean and the standard deviation for the approximation, considering an hour, are given by:

• $\mu = 60 \times 2.5 = 150$.

• $\sigma = \sqrt{150} = 12.25$.

Using continuity correction, as the Poisson distribution is discrete and the normal is continuous, P(X ≥ 125) is one subtracted by the p-value of Z when X = 124.5, hence:

$Z = \frac{X - \mu}{\sigma}$

Z = (124.5 - 150)/12.25

Z = -2.08

Z = -2.08 has a p-value of 0.0188.

1 - 0.0188 = 0.9812, hence:

P(X ≥ 125) = 0.9812.

More can be learned about the normal distribution at https://brainly.com/question/4079902

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