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

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

• A. By taking the mean of all sample outcomes
• B. By calculating the sum of all probabilities
• C. By integrating (or summing) the product of that function and its probability density/mass function
• D. By finding the median of the distribution

Answer: (C)

The expected value of a function of a random variable is calculated by integrating or summing the product of that function and the probability density function (for continuous random variables) or the probability mass function (for discrete random variables). This process helps to capture the average or mean outcome of the function based on the distribution of the random variable. For a continuous random variable, the expected value of a function \( g(X) \) is expressed mathematically as: \[ E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx \] where \( f(x) \) is the probability density function of the random variable \( X \). For a discrete random variable, the expected value is calculated using: \[ E[g(X)] = \sum_{i} g(x_i) P(X = x_i) \] where \( P(X = x_i) \) is the probability mass function evaluating the outcomes \( x_i \). This method ensures that we are weighting the values of the function according to their probabilities, which provides a comprehensive measure of the central tendency of the function with respect to the random variable's distribution. Thus, calculating the expected value through this method is essential for

The expected value of a function of a random variable is calculated by integrating or summing the product of that function and the probability density function (for continuous random variables) or the probability mass function (for discrete random variables). This process helps to capture the average or mean outcome of the function based on the distribution of the random variable.

For a continuous random variable, the expected value of a function ( g(X) ) is expressed mathematically as:

[

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

]

where ( f(x) ) is the probability density function of the random variable ( X ).

For a discrete random variable, the expected value is calculated using:

[

E[g(X)] = \sum_{i} g(x_i) P(X = x_i)

]

where ( P(X = x_i) ) is the probability mass function evaluating the outcomes ( x_i ).

This method ensures that we are weighting the values of the function according to their probabilities, which provides a comprehensive measure of the central tendency of the function with respect to the random variable's distribution. Thus, calculating the expected value through this method is essential for