The amount of snowfall in feet in a remote region of Alaska...

The amount of snowfall in feet in a remote region of Alaska in the month of January is a continuous random variable with probability density function $$f(x)=\frac{2}{9} x(3-x) \quad(0 \leq x \leq 3)$$ Find the amount of snowfall one can expect in any given month of January in Alaska. Find the variance and the standard deviation.

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Key Concepts

Standard Deviation

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the original variable. It is useful for interpreting the typical deviation from the mean, giving a more intuitive sense of the variability in the data.

Probability Density Function

A probability density function (pdf) defines the likelihood of a continuous random variable taking on different values. It must be non-negative over its domain and integrate to 1 across the entire range, ensuring that the total probability equals 1.

Expected Value (Mean)

The expected value of a continuous random variable is a measure of its central tendency, calculated by integrating the product of the variable and its pdf over its domain. It represents the long-term average outcome if the random process were repeated many times.

Variance

Variance quantifies the spread or dispersion of a continuous random variable around its mean. It is computed by integrating the squared deviation of the variable from its expected value, multiplied by its pdf, which measures the average of these squared differences.

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