Which of the following statements about the mean of a continuous random variable is true?

• A. The mean of a continuous random variable is always an integer.
• B. The mean of a continuous random variable is always equal to its median.
• C. The mean of a continuous random variable $X$ with probability density function $f\left(x\right)$ is given by ${\int }_{-\infty }^{\infty }xf\left(x\right)dx$.
• D. The mean of a continuous random variable is calculated as ${\int }_{-\infty }^{\infty }{x}^{2}f\left(x\right)dx$.

Answer: (C)

Verified step by step guidance

1. Recall that the mean (or expected value) of a continuous random variable \( X \) with probability density function \( f(x) \) is defined as the integral of \( x \) times \( f(x) \) over the entire range of \( X \). Mathematically, this is written as: \[ \int_{-\infty}^{\infty} x f(x) \, dx \]
2. Understand that the mean is a weighted average of all possible values of \( X \), where the weights are given by the probability density \( f(x) \). It is not necessarily an integer, nor does it have to equal the median.
3. The median is the value that divides the probability distribution into two equal halves, which is generally different from the mean unless the distribution is symmetric.
4. Note that the integral involving \( x^2 f(x) \) corresponds to the calculation of the second moment about zero, not the mean. This is related to variance, not the expected value.
5. Therefore, the correct formula for the mean of a continuous random variable \( X \) with pdf \( f(x) \) is: \[ \mu = E(X) = \int_{-\infty}^{\infty} x f(x) \, dx \]