| English | 意味 |
|---|---|
| random variable/ˈrændəm ˈveərɪəbl/ | 確率変数 |
| probability distribution/ˌprɒbəˈbɪlɪti ˌdɪstrɪˈbjuːʃn/ | 確率分布 |
| discrete random variable/dɪˈskriːt ˈrændəm ˈveərɪəbl/ | 離散型確率変数 |
| continuous random variable/kənˈtɪnjuːəs ˈrændəm ˈveərɪəbl/ | 連続型確率変数 |
A number that depends on chance
- A random variable 随机变量 assigns a number to each outcome of a chance process.
- Example: $X =$ the number of heads in $3$ coin flips.
- Its value is decided by randomness, so it has a probability distribution.
- Random variables let us do arithmetic with chance.
Discrete vs. continuous
- A discrete random variable 离散随机变量 takes countable, separate values ($0, 1, 2, \dots$).
- A continuous random variable 连续随机变量 can take any value in an interval (heights, times).
- Count → discrete; measure on a scale → continuous.
- This unit focuses on discrete distributions (a table of values and probabilities).
The probability distribution
- A probability distribution 概率分布 lists each value $x$ with its probability $P(x)$.
- Two rules make it valid: each $0 \le P(x) \le 1$, and all the $P(x)$ sum to $1$.
- It can be shown as a table or a bar-style graph.
- The distribution is the complete description of the random variable.
Probabilities of events
- To find $P(X \ge 2)$ or $P(X = 1)$, just add the relevant $P(x)$ values.
- "$X$ at least $2$" = $P(2) + P(3) + \cdots$.
- "$X$ between" = sum the probabilities in that range.
- Every event probability is a sum of the listed pieces.
A probability distribution must have all $P(x)$ in $[0,1]$ summing to exactly $1$ — if a table's probabilities don't total $1$, it isn't valid. And keep discrete vs. continuous straight: you count a discrete variable (number of heads) but measure a continuous one (a person's exact height) — the tools differ.
$X =$ heads in $2$ fair flips. Distribution: $P(0)=0.25,\ P(1)=0.5,\ P(2)=0.25$.
- Valid? All in $[0,1]$ and $0.25+0.5+0.25 = 1$. ✓
- $P(X \ge 1) = P(1) + P(2) = 0.5 + 0.25 = 0.75$.
- $P(X = 2) = 0.25$ — exactly two heads.
A random variable assigns a number to each outcome. A discrete one takes countable values; a continuous one takes any value in an interval. Its probability distribution lists $P(x)$ with each in $[0,1]$ summing to $1$; event probabilities like $P(X\ge 2)$ are sums of the relevant $P(x)$.
離散型確率分布
各棒グラフは P(x) を表し、棒グラフの合計は 1 となる。
コイン3回投げでの表の数は、どのような種類の確率変数か？
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
可算な個別の値（0,1,2,3）を取る — 離散的。
有効な確率分布では、確率の合計が 1 でなければならない。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
すべての P(x) は [0,1] にあり、正確に 1 に合計される。
X = コイン2回投げでの表の数：P(0)=0.25, P(1)=0.5, P(2)=0.25 です。P(X ≥ 1) を求めよ。
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
P(1)+P(2) = 0.5 + 0.25 = 0.75.
一人の人の正確な身長は、どのような種類の変数か？
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
身長はスケール上で測定される — 連続的。
確率過程の各結果に数を割り当てる法則を ___ 変数と呼ぶ。
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
それが確率変数の定義である。