The total time, measured in units of 100 hours, that a...

The total time, measured in units of 100 hours, that a teenager runs their hair dryer over a period of one year is a continuous random variable $X$ that has the density function $$ f(x)=\left\{\begin{array}{ll} x, & 0<x<1 \\ 2-x, & 1 \leq x<2 \\ 0, & \text { elsewhere } \end{array}\right. $$ Use Theorem 4.6 to evaluate the mean of the random variable $Y=60 X^{2}+48 X,$ where $Y$ is equal to the number of kilowatt hours expended annually.

The total time, measured in units of 100 hours, that a teenager runs their hair dryer over a period of one year is a continuous random variable $X$ that has the density function
$$
f(x)=\left\{\begin{array}{ll}
x, & 0<x<1 \\
2-x, & 1 \leq x<2 \\
0, & \text { elsewhere }
\end{array}\right.
$$
Use Theorem 4.6 to evaluate the mean of the random variable $Y=60 X^{2}+48 X,$ where $Y$ is equal to the number of kilowatt hours expended annually.

Key Concepts

Probability Density Function (PDF)

A probability density function describes the likelihood of a continuous random variable taking on a particular value. It is a function that, when integrated over an interval, gives the probability that the variable lies within that interval. The area under the curve of the PDF over the entire range of the variable equals one, ensuring that the total probability is normalized.

Expected Value

The expected value, or mean, of a continuous random variable is the long-run average value obtained when the variable is sampled many times. It is calculated by integrating the product of the variable and its PDF over its entire range. This concept is fundamental as it provides a measure of the central tendency of the distribution.

Transformation of Random Variables (Law of the Unconscious Statistician)

When dealing with a function of a random variable, the Law of the Unconscious Statistician allows us to compute the expected value of the transformed variable without needing to first find its PDF. This is done by integrating the function applied to the variable multiplied by the original variable’s PDF over the appropriate interval. This approach simplifies the calculation of the mean for functions of random variables.

Linearity of Expectation

The linearity of expectation states that the expected value of a sum of random variables (or functions thereof) is equal to the sum of the expected values, regardless of whether the variables are independent. This property allows for the separate calculation of the expected values of individual components in a linear combination, making the evaluation process much easier.

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