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A continuous random variable is a variable that can take any value within a given range. Unlike discrete random variables, which have distinct values, continuous variables are like variables that can move smoothly across values. Imagine a thermometer measuring temperature. It doesn’t just jump from one degree to another but smoothly transitions, capturing every possible temperature. This continuity makes continuous random variables vital for modern statistics and probability. In our exercise, the variable \(X\) is continuous since it covers all positive values \(>0\). This characteristic influences how we compute expectations because we work with integrations instead of summations.

A density function is a mathematical description of the probability distribution of a continuous random variable. This function shows us how dense or compressed the data can be at different points. In simpler terms, it tells us how likely it is for a variable to be at a particular value.

• The function must be non-negative.
• The total area under the curve of the density function must be equal to 1, which symbolizes total probability.

In our exercise, the density function \(f(x)=e^{-x}\) is provided for the random variable \(X\). Here, \(e^{-x}\) is the density function for an exponential distribution, emphasizing that lower values of \(X\) are more probable than higher values. As \(x\) increases, \(e^{-x}\) decreases, indicating a rapid drop-off in probability.

Integration is a mathematical tool used to sum up infinitesimally small data slices over a continuous range. When dealing with continuous random variables, integration becomes essential because it allows us to consider all potential values the variable might assume. It is analogous to finding the area under a curve on a graph. In our problem, determining the expected value of \(g(X)\) requires integrating the function \(e^{x/3}\) over the range \(0\) to \(+\infty\). The integration process involves simplifying the function, often using methods like substitution. This is similar to how you might change variables in an algebraic equation to make it easier to solve. Mastery of integration techniques is crucial for students to handle expected value calculations for continuous random variables effectively.

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It grows rapidly and is vital in many scientific fields, including probability. The function \(f(x)=e^{-x}\) is a typical exponential function; its rapid decrease makes it apt for modeling decay processes, like radioactive decay or discharging capacitors.

• Exponential functions have unique properties: they never go negative, and their derivatives and integrals are proportionate to the maximum value of the function.
• In probability, exponential functions often express waiting times or life spans of systems.

In this exercise, the exponential function \(g(X)=e^{2X/3}\) modifies the behavior of the density function \(e^{-x}\), creating a new composite exponential that we ultimately integrate. The interplay between \(e^{2X/3}\) and \(e^{-x}\) showcases how different exponential functions can come together to model real-world probability scenarios.