For the PDF p(x)=(1)/(2) e^-|x|, determine the mean, the...

For the PDF $p(x)=\frac{1}{2} e^{-|x|}$, determine the mean, the variance, and the probability that a random variable will fall within $\pm \sigma_x$ of the mean value.
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Figure 3.22. An example PDF.

Key Concepts

Laplace Distribution

The Laplace distribution, also known as the double exponential distribution, is characterized by a probability density function that decays exponentially with the absolute value of the variable. It is symmetric about its mean, which allows for simplification in calculations of central measures and spread, and its heavier tails compared to the normal distribution affect probability calculations at distances from the mean.

Mean of a Distribution

The mean, or expected value, is a measure of central tendency that represents the average outcome of a random variable weighted by its probability density. For symmetric distributions like this Laplace distribution, the mean typically lies at the center of the distribution, making it a natural point of reference for measuring deviation.

Variance

Variance is a statistical measure that quantifies the dispersion of data points around the mean. It is calculated as the expected value of the squared deviations from the mean and provides insight into the spread of the distribution. For the Laplace distribution, knowing the variance is crucial when evaluating the spread relative to the central value.

Probability Integration

Probability integration involves computing the likelihood that a random variable falls within a particular interval by integrating the probability density function over that interval. This concept is central in statistics for determining probabilities such as the chance that a variable lies within one standard deviation of the mean, which assesses concentration around the center.

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