How is the expected value of a continuous random variable calculated?

How is the expected value of a continuous random variable calculated?

• A. As the sum of all possible values times their probabilities
• B. As the integral of the random variable over its probability mass function
• C. As the integral of the product of the random variable and its probability density function
• D. As the average value of the random variable over time

Answer: (C)

The expected value of a continuous random variable is calculated using an integral that involves the product of the random variable and its probability density function (PDF). This process captures the idea that the expected value is not only influenced by the values of the random variable but also by how likely each value is to occur. Specifically, the expected value \( E[X] \) of a continuous random variable \( X \) with probability density function \( f(x) \) is defined as: \[ E[X] = \int_{-\infty}^{\infty} x \cdot f(x) \, dx \] This integral sums up all possible values of \( x \), weighted by their likelihood as indicated by the PDF \( f(x) \). The integral encompasses the entire range of the variable, accounting for the continuous nature of the distributions. This method distinguishes itself from the calculation of the expected value for discrete random variables, where the expected value is found as the sum of each outcome multiplied by its probability. In the context of continuous random variables, using the probability mass function is irrelevant because continuous variables do not have a probability mass function; instead, they utilize a probability density function. Thus, the correct method for calculating the expected value for continuous random

The expected value of a continuous random variable is calculated using an integral that involves the product of the random variable and its probability density function (PDF). This process captures the idea that the expected value is not only influenced by the values of the random variable but also by how likely each value is to occur.

Specifically, the expected value ( E[X] ) of a continuous random variable ( X ) with probability density function ( f(x) ) is defined as:

[ E[X] = \int_{-\infty}^{\infty} x \cdot f(x) , dx ]

This integral sums up all possible values of ( x ), weighted by their likelihood as indicated by the PDF ( f(x) ). The integral encompasses the entire range of the variable, accounting for the continuous nature of the distributions.

This method distinguishes itself from the calculation of the expected value for discrete random variables, where the expected value is found as the sum of each outcome multiplied by its probability. In the context of continuous random variables, using the probability mass function is irrelevant because continuous variables do not have a probability mass function; instead, they utilize a probability density function. Thus, the correct method for calculating the expected value for continuous random