The amount of rainfall (in inches) on a tropical island in...

The amount of rainfall (in inches) on a tropical island in the monthof August is a continuous random variable with probability density function $$f(x)=\frac{1}{36} x(6-x) \quad(0 \leq x \leq 6)$$ What is the expected amount of rainfall on any day in August on that island? Find the variance and standard deviation.

The amount of rainfall (in inches) on a tropical island in the monthof August is a continuous random variable with probability density function $$f(x)=\frac{1}{36} x(6-x) \quad(0 \leq x \leq 6)$$
What is the expected amount of rainfall on any day in August on that island? Find the variance and standard deviation.

Key Concepts

Standard Deviation

The standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the original variable. It quantifies the typical distance of the random variable's values from the expected value, offering an intuitive sense of variability in the data.

Continuous Random Variable

A continuous random variable is one that can take on any value within a certain interval or range. In such cases, probabilities are determined not at specific points but over intervals, and are computed using an integration of the probability density function over that interval.

Probability Density Function

The probability density function (pdf) describes how the probabilities of a continuous random variable are distributed. It must be non-negative and its integral over the entire space equals 1. The pdf allows us to calculate probabilities for intervals by integrating the function over those intervals.

Expected Value

The expected value of a continuous random variable is the mean or average outcome over many trials. It is calculated by integrating the product of the variable and its probability density function over its range, providing a measure of the center of the distribution.

Variance

Variance measures the spread or dispersion of the random variable's values around the expected value. It is computed as the expected value of the squared deviation from the mean, which is found by integrating the squared difference between the variable and its mean over the probability distribution.

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