Random Variables and the Normal Distribution (Statistics 1)

Discrete random variables and their expectation, the binomial and geometric distributions, the normal distribution and its standardisation, and when a binomial may be approximated by a normal.

หลักสูตรแกนกลาง / สสวท.: สถิติและความน่าจะเป็น (Statistics and Probability)ตรงกับ ม.6 การแจกแจงปกติ (เพิ่มเติม) — การแจกแจงทวินามและเรขาคณิตอยู่ในระดับเดียวกันของหลักสูตรไทย
Common Core (US): S-MD, S-IC
Cambridge: 9709 Probability & Statistics 1 — Discrete random variables; The normal distribution

14.1 Discrete random variables

Expectation and variance

E (X) = \sum x P (X = x) Var (X) = \sum x^{2} P (X = x) - [E (X)]^{2}

The probabilities in a distribution table must add to $1$ — which is usually how an unknown in the table is found.

ตัวอย่าง

A random variable $X$ takes the values $1, 2, 3$ with probabilities $0.2$ , $0.5$ and $0.3$ . Find $E (X)$ and $Var (X)$ .

วิธีทำ

1. $E (X) = 0.2 + 1.0 + 0.9 = 2.1$
2. $E (X^{2}) = 0.2 + 2.0 + 2.7 = 4.9$
3. $Var (X) = 4.9 - 2. 1^{2}$

ตอบ $E (X) = 2.1$ and $Var (X) = 0.49$

ข้อควรระวัง

$E (X^{2})$ is not $[E (X)]^{2}$ . Squaring the values and then averaging is a different operation from averaging and then squaring, and the whole variance formula is the gap between them.

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