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A random variable is a quantity that can take on several different values, depending on chance. Examples include:
If $X$ is a continuous random variable, then the probability of any one outcome is zero. Instead, we consider the probability of a range of outcomes. The probability density function (abbreviated `pdf') $f_X(x)$ gives the probability per unit length. In the following video, we show how to use such a function, and we learn about three standard examples, called the uniform, exponential, and normal distributions.
Of course, there is a 100% chance of $X$ taking on some value, so $$\int_{\infty}^\infty f_X(x) dx = 1.$$ If $X$ is restricted to a smaller range than $(\infty, \infty)$, we just integrate over that range.
If $X$ is a random variable, the expectation or mean value of $X$ is the average value we get when we run the experiment over and over again. In the following video, we explain how to use integrals to compute expectations, variances, and standard deviations.
If the mean value of $X$ is $\mu$, then the variance of $X$ is the average value of $(X\mu)^2$. This is a measure of how wide the probability distribution is, and has units of (length)${}^2$, and is denoted $Var(X)$ or $\sigma^2$.
