Random variable
Definition taught in Random variables and distributions, part of Probability
Random variable
| Type | Definition |
| Taught in | Random variables and distributions |
| Part | Probability |
| Properties | 1 |
| Questions using it | 4 |
A random variable is a function X from the sample space \Omega to the real numbers. The event \{X\le x\} is the set of outcomes \omega with X(\omega)\le x.
Properties
Expectation of a function
Main article: Expectation of a function
If g is a real function and X is discrete with mass function f, then \mathbb{E} g(X)=\sum_x g(x)\,f(x). If X is continuous with density f, then \mathbb{E} g(X)=\int_{-\infty}^{\infty} g(x)\,f(x)\,\,\mathrm{d} x, whenever the sum or integral converges absolutely.
In the discrete case, group the outcomes on which g takes the value y. The mass at y is \sum_{x:g(x)=y}f(x), so \mathbb{E} g(X)=\sum_y y\sum_{x:g(x)=y}f(x)=\sum_x g(x)\,f(x).
Used in
- Taught in Random variables and distributions, where the derivation and examples are.
- Also used in Joint distributions · Functions of random variables and limit theorems · Reliability · Statistical inference · Linear regression
- Questions that use it (4) — A cubic distribution function · Three-point expectation · Recovering the second moment · A shifted exponential density
- All 27 questions on Random variables and distributions
See also
Sample space · Independent random variables · Joint probability density · Joint probability mass function · Memoryless property · Cumulative distribution function · Discrete distribution · Continuous distribution