22 Lecture 21, March 01, 2024
22.1 Law of unconciousness of statistician, continuous version
Definition 22.1 (LOTUS) If \(X\) is a continuous random variable with pdf \(f(x)\), and \(g:\mathbb{R}\to \mathbb{R}\) is a function, then \[ \mathbb{E}g(x) = \int_{-\infty}^\infty g(x)f(x)dx \] provided the expression exists.
By above, we can calculate the expectation and the variance as follows
\(\mathbb{E}X = \int_{-\infty}^\infty x f(x) dx\)
\(\mathbb{V}ar(X) = \mathbb{E}[(X-\mathbb{E}X)^2] = \int_{-\infty}^\infty (x-\mathbb{E}X)^2 f(x)dx\).
Similar to the discrete random variable case, we have the shortcut formula to calculate the variance: \[ \mathbb{V}ar(X) = \mathbb{E}[X^2] - (\mathbb{E}X)^2. \]
22.2 function of random variable
- If \(X\) is a random variable and \(g\) is a function, then \(Y=g(X)\) is also a random variable
- By the law of the unconscious statistician,
\[ \mathbb{E}(Y)=\mathbb{E}(g(X)) = \begin{cases} \sum_{\text{all }x} g(x)\cdot f(x),\quad&\text{if $X$ discrete with pf }f,\\ \int_{\mathbb{R}} g(x)\cdot f(x)dx,\quad&\text{if $X$ continuous with pdf }f\end{cases} \]
- Next, we are studying how to find the distribution of \(Y=g(X)\).