Continuous random variables · 连续随机变量
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| continuous random variable/kənˈtɪnjuːəs ˈrændəm ˈveərɪəbl/ | 连续型随机变量 | lián xù xíng suí jī biàn liàng |
| probability density function/ˌprɒbəˈbɪlɪti ˈdensɪti ˈfʌŋkʃn/ | 概率密度函数 | gài lǜ mì dù hán shù |
| area/ˈeərɪə/ | 面积 | miàn jī |
| cumulative distribution function/ˈkjuːmjʊlətɪv ˌdɪstrɪˈbjuːʃn ˈfʌŋkʃn/ | 累积分布函数 | lěi jī fēn bù hán shù |
The curve that tells probabilities
- A continuous random variable 连续型随机变量 can take any value in a range — like the exact time a bus arrives, or the precise height of a person.
- Unlike discrete variables, you can't list probabilities for each value. Instead, a probability density function 概率密度函数 (pdf) describes the shape, and probabilities are areas 面积 under the curve.
讲概率的曲线
- 一个连续随机变量能取一个范围内的任何值——像一辆公交到达的确切时间,或一个人的精确身高。
- 不像离散变量,你不能为每个值列出概率。取而代之,一个概率密度函数(probability density function,pdf)描述形状,而概率是曲线下的面积。
The probability density function
- A continuous random variable can take any value in a range, described by a probability density function $f(x)$:
Worked example. $f(x) = \dfrac{x}{8}$ for $0 \leq x \leq 4$, zero otherwise. Check: $\int_0^4 \dfrac{x}{8}\,dx = \left[\dfrac{x^2}{16}\right]_0^4 = \dfrac{16}{16} = 1$. ✓
$f(x)$ is NOT a probability. The value of $f(x)$ at a point can be greater than 1. Only the area under the curve between two points gives a probability.
For a continuous variable, probability is the area under the density f(x)
概率密度函数
- 一个连续随机变量能取一个范围内的任何值,由一个概率密度函数 $f(x)$ 描述:
算例。 $f(x) = \dfrac{x}{8}$,$0 \leq x \leq 4$,其他为零。检查:$\int_0^4 \dfrac{x}{8}\,dx = \left[\dfrac{x^2}{16}\right]_0^4 = \dfrac{16}{16} = 1$。✓
$f(x)$ 不是一个概率。 $f(x)$ 在一个点的值可以大于 1。只有曲线下两点之间的面积给出一个概率。
对一个连续变量,概率是密度 f(x) 下的面积
Area = probability · 面积 = 概率
P(a < X < b) = ∫ f(x) dx
For a continuous variable, probability · 概率 is the area · 面积 under the density curve between two values. · 对一个连续变量,概率是密度曲线下两个值之间的面积。
A probability density function must satisfy ∫f(x)dx over all x equals what value? · 一个概率密度函数必须满足在所有 x 上 ∫f(x)dx 等于什么值?
The total area under a pdf is always 1. · 一个 pdf 下的总面积总是 1。
The value of f(x) at a point is the probability that X equals that value. · f(x) 在一个点的值是 X 等于那个值的概率。
f(x) is a density, not a probability. Only the area under f gives a probability. f(x) can even exceed 1. · f(x) 是一个密度,不是一个概率。只有 f 下的面积给出一个概率。f(x) 甚至可以超过 1。
For f(x) = x/8 on 0 ≤ x ≤ 4, verify ∫₀⁴ (x/8) dx = [x²/16]₀⁴. What is the result? · 对 0 ≤ x ≤ 4 上的 f(x) = x/8,验证 ∫₀⁴ (x/8) dx = [x²/16]₀⁴。结果是多少?
[x²/16]₀⁴ = 16/16 − 0 = 1. The total area is 1, confirming it is a valid pdf. · [x²/16]₀⁴ = 16/16 − 0 = 1。总面积是 1,确认它是一个有效的 pdf。
Finding probabilities
- A probability is an area under $f$: $P(a < X < b) = \displaystyle\int_a^b f(x)\,dx$.
Probabilities are areas under f(x); the total area over all x must equal 1.
求概率
- 一个概率是 $f$ 下的一个面积:$P(a < X < b) = \displaystyle\int_a^b f(x)\,dx$。
概率是 f(x) 下的面积;在所有 x 上的总面积必须等于 1。
For a continuous variable, P(a < X < b) is the area under f(x) between a and b. · 对一个连续变量,P(a < X < b) 是 f(x) 下 a 和 b 之间的面积。
Probabilities for a continuous variable are areas under the density function. · 一个连续变量的概率是密度函数下的面积。
Finding the mean and variance
- The mean is $E(X) = \displaystyle\int_{-\infty}^{\infty} x\,f(x)\,dx$.
- The variance is $\text{Var}(X) = E(X^2) - (E(X))^2$, where $E(X^2) = \displaystyle\int_{-\infty}^{\infty} x^2\,f(x)\,dx$.
求平均数和方差
- 平均数是 $E(X) = \displaystyle\int_{-\infty}^{\infty} x\,f(x)\,dx$。
- 方差是 $\text{Var}(X) = E(X^2) - (E(X))^2$,其中 $E(X^2) = \displaystyle\int_{-\infty}^{\infty} x^2\,f(x)\,dx$。
For f(x) = ½x on 0 ≤ x ≤ 2, E(X) = ∫₀² x(½x) dx = [x³/6]₀². What is E(X)? (≈, 2 dp) · 对 0 ≤ x ≤ 2 上的 f(x) = ½x,E(X) = ∫₀² x(½x) dx = [x³/6]₀²。E(X) 是多少?(≈,2 位小数)
E(X) = ∫₀² ½x² dx = [x³/6]₀² = 8/6 = 4/3 ≈ 1.33. · E(X) = ∫₀² ½x² dx = [x³/6]₀² = 8/6 = 4/3 ≈ 1.33。
The cumulative distribution function 累积分布函数
- The cdf $F(x) = P(X \leq x) = \displaystyle\int_{-\infty}^x f(t)\,dt$.
- $F(x)$ is the area under $f$ up to $x$. It always increases from 0 to 1.
- Use the density function to find the median and other percentiles.
累积分布函数
- cdf $F(x) = P(X \leq x) = \displaystyle\int_{-\infty}^x f(t)\,dt$。
- $F(x)$ 是 $f$ 下直到 $x$ 的面积。它总是从 0 增加到 1。
- 用密度函数求中位数(median)和其他百分位数(percentiles)。
The cumulative distribution function F(x) gives: · 累积分布函数 F(x) 给出:
F(x) = P(X ≤ x) = the area under f up to x. · F(x) = P(X ≤ x) = f 下直到 x 的面积。
You've got it
- a pdf has $f(x) \geq 0$ and total area $\int f = 1$
- a probability is the area under $f$ between the limits
- the mean $E(X) = \int x\,f(x)\,dx$
你掌握了
- 一个 pdf 有 $f(x) \geq 0$ 和总面积 $\int f = 1$
- 一个概率是 $f$ 下两个限之间的面积
- 平均数 $E(X) = \int x\,f(x)\,dx$