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Chapter 3: Problem 12

Compute the measures of skewness and kurtosis of the Poisson distribution with mean \(\mu\).

Short Answer

Expert verified
The skewness and kurtosis of a Poisson distribution with mean \( \mu \) are given by \( \mu^{-0.5} \) and \( \frac{1}{\mu} \) respectively.

Step by step solution

01

Compute the skewness

Use the standard formula for the skewness of a Poisson distribution. This formula is: \( \mu^{-0.5} \). Plug in the value of \( \mu \) and evaluate the expression to obtain the skewness.
02

Compute the kurtosis

Utilize the standard formula for the kurtosis of a Poisson distribution. The formula is: \( \frac{1}{\mu} \). Substitute the given value of \( \mu \) into the formula and compute the value to get the kurtosis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Skewness
Skewness is a measure of the asymmetry of a probability distribution around its mean. It can tell us whether the distribution is skewed to the left, skewed to the right, or symmetric. In a symmetric distribution, the mean, median, and mode are equal. However, when a distribution is skewed:
  • If skewness is negative, the distribution is skewed to the left, with a longer left tail.
  • If skewness is positive, the distribution is skewed to the right, with a longer right tail.
  • If skewness is zero, the distribution is perfectly symmetric.
The Poisson distribution has a skewness formula given by \( rac{1}{ ext{mean}^{0.5}} \). This formula indicates that the skewness of a Poisson distribution depends on its mean, \( ext{mean} \).
When the mean is larger, the skewness decreases, implying the distribution becomes more symmetric as the mean increases.
Exploring Kurtosis
Kurtosis helps describe the shape of a distribution's tails in relation to its overall shape. A key idea behind kurtosis is understanding how outliers affect the distribution.
  • High kurtosis indicates heavier tails, showing more outliers.
  • Low kurtosis indicates lighter tails and fewer outliers.
  • A normal distribution has a kurtosis approximately equal to 3.
For the Poisson distribution, the kurtosis is calculated using the formula \( \frac{1}{ ext{mean}} \). This reflects that kurtosis inversely relates to the mean. As the mean increases, the distribution tends towards a lower kurtosis, suggesting it appears flatter at its peak and has thinner tails, becoming more normally distributed.
Moment Generating Functions in Poisson Distribution
Moment Generating Functions (MGFs) provide a powerful tool in probability and statistics for handling distributions. The MGF of a random variable is defined as\( M(t) = E[e^{tX}] \). It's a useful technique for finding moments of a probability distribution, such as mean, variance, etc.
  • The MGF for a Poisson distribution with mean \( \mu \) is given by \( M(t) = e^{ ext{mean}(e^{t} - 1)} \).
  • This MGF neatly encapsulates all moments of the distribution. By differentiating \( M(t) \) and evaluating it at \( t=0 \), the moments such as mean and variance can be derived efficiently.
Using this function, statisticians can easily compute how various characteristics of the distribution originate and interact, giving insights into its behavior under different conditions.

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