Find the expected value, the variance, and the standard...

Find the expected value, the variance, and the standard deviation, when they exist, for each probability density function.
$$f(x)=\left\{\begin{array}{ll}{\frac{x^{3}}{12}} & {\text { if } 0 \leq x \leq 2} \\ {\frac{16}{3 x^{3}}} & {\text { if } x>2}\end{array}\right.$$

Key Concepts

Standard Deviation

The standard deviation is the positive square root of the variance and provides a measure of dispersion in the same units as the variable itself. It offers an intuitive understanding of how spread out the observations are around the mean. Like the variance, its existence is tied to the convergence of the necessary integrals, making it a fundamental tool in statistical analysis.

Variance

Variance quantifies the spread or dispersion of a random variable around its mean (expected value). It is defined as the expected value of the squared deviation from the mean, commonly calculated by subtracting the square of the expected value from the expected value of the square of the random variable. The variance exists only if the corresponding integrals converge, and it is essential for understanding the variability inherent in the distribution.

Probability Density Function

A probability density function (pdf) describes the likelihood of a continuous random variable taking on a given value. It must be nonnegative for all values and integrate to one over its entire range to represent a valid probability model. In problems involving a pdf, piecewise definitions may occur, and each segment needs to be correctly integrated over its respective interval to ensure normalization and accurate calculations of subsequent statistical measures.

Expected Value

The expected value is a measure of the central tendency of a continuous random variable and is computed as the integral of the product of the variable and its probability density function over the full range of the variable. This weighted average provides insight into the 'center' of the distribution, and its existence requires that the integral converges. It plays a crucial role in moments and further statistical analysis.

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