Find the mean, variance, and standard deviation of the...

Find the mean, variance, and standard deviation of the random variable $x$ associated with the probability density function over the indicated interval.
$$f(x)=\frac{1}{9} x e^{-x / 3} ;[0, \infty)$$

Key Concepts

Moment Calculations

Moment calculations involve integrating powers of the random variable with respect to its probability density function. These calculations, including the first moment (mean) and the second central moment (variance), are critical tools for understanding the distribution's shape, spread, and other characteristics, such as skewness and kurtosis.

Variance

Variance quantifies the spread or dispersion of a random variable's values around the mean. It is computed as the expected value of the squared deviations from the mean, which involves integrating the squared difference (between the variable and its mean) weighted by the probability density function. This measure is central in assessing how much the values differ from the average.

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the random variable. This metric facilitates a more intuitive understanding of the spread of the distribution, indicating how much the values typically deviate from the mean.

Expected Value

The expected value, or mean, measures the central tendency of a random variable's distribution. It is calculated by integrating the variable multiplied by its probability density function over the entire domain. This concept provides a measure of the 'center' or average outcome one can expect from the distribution.

Probability Density Function

The probability density function (pdf) defines the likelihood of a continuous random variable taking a specific value. It is non-negative across its domain and integrates to one over the whole range, serving as the basis for all subsequent probability calculations in continuous probability distributions.

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