1.5 Random Variables

Definition 1.5.1

A function X is called random variable1, if it is assign to each element \(C\in\mathscr{C}\) one and only one number \(X\left(c\right)\in x\). The space or range of X is \(D=\left\{x:x=X\left(c\right), \ C\in\mathscr{C}\right\}\).

  • The random variable is called discrete random variable if D is contable sequence.
  • The random variable is called continuous random variable if D is interval of real number.
  • Note that the probability function P is defined on \(\beta\).
  • We define a probability function \(p_x\) defined on F and \(p_x\) is called induced probability function.

ex 1.5.1) Toss two fair dice.

  • X : Sum of up faces
  • \(\mathscr{C} \ : \ \left\{\left(1,1\right),\left(1,2\right),\cdots,\left(6,6\right)\right\}\) .
  • \(D=\left\{2,3,\cdots,12\right\}\).

Find the probability of sum 4.

\[\begin{aligned} P\left(\left(1,3\right)\cup\left(2,2\right)\cup\left(3,1\right)\right)\\ = \frac{3}{36} = p_X\left(4\right) = P\left(4\right) \end{aligned}\]

Definition 1.5.2 (Cumulative Distribution Function)

The cdf of random variable X is

\[\begin{aligned} F_X\left(x\right) = P\left(X\le x\right) =&\ P\left(c: X\left(c\right)\le x\right)\\ =&\ P_X\left(\left[-\infty, x\right]\right) \end{aligned}\]

Ex 1.5.3)

​ X: upface of tossing a fair dice.

​ Find CDF of X

Ex 1.5.4)

​ X: real number chosen at random in (0, 1).

\[\begin{aligned} p_X\left(\left(a, b\right)\right) =& \left(b-a\right)\\ \\ F_X\left(x\right) = P\left(X\le x\right)\\ = &\ 0,\quad \text{if }x\lt 0\\ =&\ x, \quad \text{if } 0\le x \le 1\\ =&\ 1, \quad \text{if } x\ge 1 \end{aligned}\]
Theorem 1.5.2 (Property of CDF)
  1. \(\forall a,b,\ a\lt b\Rightarrow F\left(a\right)\le F\left(b\right)\).
  2. \(\lim_{x\rightarrow-\infty}F\left(x\right)=0\).
  3. \(\lim_{x\rightarrow\infty}F\left(x\right)=1\).
  4. \(\lim_{x\leftarrow x_0}=F\left(x_0\right)\). (right continuous)

c.f)

  • \(x\leftarrow x_0\), convergence from the right.

  • \(x\rightarrow x_0\), convergence from the left.

    \[\begin{aligned} \\ \lim_{x\leftarrow x_0}F\left(x\right)&\ :\text{Righthand side limit}\\ \lim_{x\rightarrow x_0}F\left(x\right)&\ :\text{Lefthand side limit}\\ \end{aligned}\]
proof.
\[\begin{aligned} \text{1.}&\ \quad F\left(a\right) =\ P\left(X\le a\right), \quad a\lt b,\ \ \forall a,b\\ &\ \left\{X\le a\right\}\subset \left\{X\le b\right\}, \quad b\gt a\\ &\Rightarrow P\left(X\le a\right)\le P\left(X\le b\right)\ \ \text{by theorem 1.3.3} \\ \\ \text{2.}&\ \lim_{x\to -\infty}\left\{X\le x\right\}= \emptyset \\ &\ \lim_{x\to -\infty}P\left(X\le x\right) = P\left(\emptyset\right) = 0 \\ \\ \text{3.}&\ \lim_{n\rightarrow \infty}\left\{X\lt x\right\} = \mathscr{C}\\ &\ \lim_{x\to \infty}\left\{X\le x\right\} = P\left(\mathscr{C}\right) = 1 \end{aligned}\]

4) Let \(\left\{X_n\right\}\) be a sequence st. \(x\leftarrow x_0\) and let \(C_n=\left\{X\le x_n\right\}\). Then \(C_n\) is decreasing set and \(\bigcap_{n=1}^\infty C_n=\left\{X\le x_0\right\}\). \(P\left(\bigcap_{n-1}^{\infty}C_n\right)=P\left(X\le x_0\right)=F(x_0)=\lim_{n\rightarrow\infty}P(C_n)=\lim_{n\rightarrow}F(x_n)\).

Theorem 1.5.2
\[\begin{aligned} P(a\lt x \le b) = F_X\left(b\right) - F_X(a),\quad \forall a\lt b \\ \left(-\infty, b\right] = \left(-\infty, a\right]\cup \left(a, b\right]\\ F_X\left(b\right) = F_X\left(a\right) + P\left(a\lt x \le b\right) \\ \therefore P(a\lt x \ le b) = F_X(b) - F_X(a) \end{aligned}\]
Theorem 1.5.3
  • \(P(X=x)=F_X(x)-F_X(x^-)\), where \(F_X(x)\) : Left hand limit at x
proof.

  1. r.v. ↩