1. Random Variable (= r.v.)¶

1.1. Definition¶

(Rough) Definition: Variable with a probability

Probability that $x=a$

$$\triangleq \; P_X(x=a)\;= \; P(x=a) \;\implies\; \begin{cases} \text{1)} \; P(x=a) \geq 0 \\ \text{2)} \; \sum\limits_{\text{all}} P(x)=1 \end{cases}$$

$\begin{cases} \text{continuous r.v.} \qquad \text{if} \;x \;\text{is continuous}\\ \text{discrete r.v.} \quad \qquad \, \text{if} \;x \;\text{is discrete} \end{cases}$

Example

$x$: die outcome

$$ P(x=1)=P(x=2)= \;\dotsb \;= P(x=6) = \frac{1}{6} $$

Question

$$ \begin{align*} y &= x_1 + x_2: \;\;\; \text{ sum of two dice} \\\\ P_Y(y=5) &= \text{?} \end{align*} $$