[P&R] 02. Random Variable(1)

🪄 Definition

• A random variable $X(u)$ is a mapping from the sample space to the real line.
  In other words, an assginment of a number to every possible outcomes.

  $X(u) : U \rightarrow R$

 

⭐ We can have several random variables defined on the same sample space

     - Discrete RV

$X(f_i) = 10i,$ in die experiment
$X \in \{10, 20, 30, 40, 50, 60\}$ ~ Discrete RV

• $case1$ $f_2 = f_3 = 20$ is okay.
• $case2$ $f_2 = 20, f_2 = 30$ is not function.

     - Continuous RV

$U = (0, 12 ],$ $X(u) = u^2,$ $X \in (0, 144]$ ~ Continuous RV

---

■ Probability Mass Function a.k.a. PMF

• $X(u) \rightarrow$ When we define the RV, describe what kind of experiment it was.
  In PMF, we describe the RV, all we have to give is Probability.

  $P_X(x) = P\{X = x\},$ each of elements have probability

  • $X :$ Random Variable
  • $x :$ Any number of set

 

• Example of Binomial random variable
       • $X$ ~ Number of head in n independent coin tosses

  $P_X(x) =$ $n \choose x$$\times p^x \times p^{n-x}$ ~ Binomial PMF

• Example of Geometric random variable
       • $X$ ~ Number of coin tosses until the fist head

  $P_X(x) = (1-p)^{x-1} \times p$
  

 

📙Note

• $P_X(x) \ge 0$
• $\sum_{i=x} P_X(x) = 1$

---

■ Probability Density Function a.k.a. PDF

• A continuous RV $X$ is described by a probability density function, $f_X(x)$

  • $\int\limits_{-\infty}^\infin$$f_X(x)$ $dx = 1$
  • $\int\limits_{b}^a$$f_X(x)$ $dx = P[a \le x \le b]$
  • $P(X \in B) = \int\limits_Bf_x(x)$ $dx$
  • $P(X \le X \le x + \delta) = \int\limits_x^{x+ \delta}f_X(x)$ $dx$
    If $\delta$ is very small, $\delta = 0 \rightarrow$ $P(X=x)$

---

■ Probability Distribution Function (Cumulative Distribution Function, a.k.a. CDF)

$F_X(x) = P[X \le x] = \int\limits_{-\infin}^x f_X(\theta)$ $d\theta \\f_X(x) = \frac{d}{dx}F_x(x) \rightarrow$ Find to the density function using the CDF

---

■ Properties of Probability Distribution Function

1. $F_X(\infin) = P[X \le \infin] = 1$
   $F_X(-\infin) = P[X \le -\infin] = 0$
2. $F_X(x)$ is a non-decreasig function of $x$
   In other words, if $x_1 < x_2,$ then $F_X(x_1) \le F_X(x_2)$
3. $P[X >x] = 1 - F_X(x)$
4. $F_X(x)$ is right-continuous
5. $P[x_1 < X \le x_2] = F_X(x_2) - F_X(x_1)$

   5. Proof ) $\\ \{X \le x_2 \} = \{x_1 < X \le x_2\} \bigcup \{X \le x_1\}$ $\\P\{X \le x_2\} = P\{x_1 < X \le x_2\} +P\{X \le x_1\}$ (∵ 상호배반) $\\F_X(x_2) = P\{x_1 < X \le x_2\} + F_X(x_1)$ $\\∴ P[x_1 < X \le x_2] = F_X(x_2) - F_X(x_1)$
      
      

6. $P[X=x] = P[x^- < X \le x] = F_X(x) - F_X(x^-)$

 

✏️ Example

Question. What is the $F_X(x) =$ $?$

According to the definition, $F_X(x) = \int\limits_{-\infin}^x f_x(\theta)$ $d\theta$
Then,

• $x < 1 \rightarrow F_x(x) = 0$
• $1 \le x < 2 \rightarrow F_x(x) = \int\limits_1^x (\theta-1)$ $d\theta = \frac{x^2}{2} - x + \frac{1}{2}$
• $2 \le x < 3 \rightarrow F_X(x) = \frac{1}{2} + \int\limits_2^x(3-\theta)$ $d\theta = \frac{-x^2}{2} + 3x - \frac{7}{2}$
• $x \ge 3 \rightarrow F_X(x) = 1$

---

■ Typical Density Function

• $Q(x) = 1 - G(x)$

✏️ Example

• Let's suppose that $X$ ~ $(-3, 2^2)$
  $P[|X + 3| < 2] = P\{-5 \le X \le -1\}$
  $\Leftrightarrow F_X(-1)-F_X(-5) \\ \Leftrightarrow G(\frac{-1 + 3}{2}) - G(\frac{-5+3}{2}) \\ \Leftrightarrow G(1) - G(-1) \\ \Leftrightarrow G(1) - (1-G(1)) \\ \Leftrightarrow 2 \times G(1) - 1 \\ ∴ 1 - 2 \times Q(1)$

---

■ Exponential

$f_X(x)$ of Exponential

• $\lambda \times e^{-\lambda x},$ $x \ge 0$
• $0,$ otherwise
  
  $F_X(x) = \int\limits_0^x \lambda e^{-\lambda\theta}$ $d\theta$
          $= [-e^{-\lambda\theta}]_0^x$
          $= 1 - e^{\lambda x},$ $x \ge 0$
  

---

■ Poisson

• $P_X(k) =$ $n \choose k$$\times p^k \times q^{n-k},$ At that time $\big( p = \frac{\tau}{T}, q = 1 - \frac{\tau}{T}\big)$
• Assume $T \rightarrow \infin,$ $n \rightarrow \infin,$ while $n/T \rightarrow \lambda$
  ($n/T$ means that "expected number of points in length 1")
• $P_X(k) ≃ e^{-\lambda \tau \times \frac{(\lambda \tau)^k}{k!} }$ ~ Poission RV

🪄 Question : What is the density of distance between adjacent points?

$F_Y(y) = P[Y \le y]$
$\Leftrightarrow P[Y > y] = P\{$ No Point in Interval with Length y $\}$
$\Leftrightarrow 1-F_Y(y) = e^{-\lambda y}$
$\Leftrightarrow F_Y(y) = 1 - e^{-\lambda y}$
$∴ f_Y(y) = \lambda\times e^{-\lambda y},$ $y \ge 0$

---

본 글은 HGU 2023-2 확률변수론 이준용 교수님의 수업 필기 내용을 요약한 글입니다.