Mean and Variance of a Random Variable

Mean (Expected Value)

The mean or expected value of a random variable $X$ is a measure of the central tendency of the distribution of $X$. It represents the "average" value you would expect if you were to observe the random variable many times.

For a discrete random variable $X$ with possible values $x_1, x_2, \ldots, x_n$ and corresponding probabilities $p_1, p_2, \ldots, p_n$, the mean (expected value) is calculated as:

$\mu = \mathbb{E}[X] = \sum_{i=1}^{n} x_i \cdot p_i$

For a continuous random variable $X$ with probability density function (pdf) $f(x)$, the mean is calculated as:

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

Variance

The variance of a random variable $X$ measures the spread or dispersion of its possible values around the mean. It quantifies how much the values of $X$ deviate from the mean on average.

The variance $\sigma^2$ is defined as:

$\sigma^2 = \text{Var}(X) = \mathbb{E}[(X - \mu)^2]$

For a discrete random variable, this is:

$\sigma^2 = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot p_i$

For a continuous random variable, it is:

$\sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) \, dx$

Expected Value of a Function of a Single Random Variable

If you have a random variable $X$ and you want to find the expected value of a function $g(X)$, it is calculated as follows:

For a discrete random variable $X$:

$\mathbb{E}[g(X)] = \sum_{i=1}^{n} g(x_i) \cdot p_i$

For a continuous random variable $X$:

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

This generalizes the concept of expected value to any function of the random variable, not just the random variable itself.

Summary

• Mean (Expected Value): A measure of central tendency.
• Variance: A measure of how spread out the values of the random variable are around the mean.
• Expected Value of a Function: Calculated by taking the weighted average (for discrete) or integral (for continuous) of the function applied to the random variable.

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Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. How is the covariance between two random variables calculated?
2. What is the significance of the standard deviation in relation to variance?
3. How do we calculate the expected value of a random variable raised to a power, say $\mathbb{E}[X^2]$?
4. Can the variance of a random variable ever be negative? Why or why not?
5. What is the law of large numbers and how does it relate to the mean of a random variable?

Tip: The expected value gives you the long-term average of a random variable, but the variance tells you how much the values typically deviate from this average. Understanding both is key in probability and statistics.