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机器产量方差:25 人总产量标准差

Machine Variance

专题
Probability / 概率
难度
L2

题目详情

一家披萨店有 25 名独立工人。每位工人每天制作披萨数量的标准差为 60。

问:全店每天披萨总产量的标准差是多少?

A pizza shop consists of 25 independent workers that make pizzas. The standard deviation of the number of pizzas produced by each worker in a day is 60. Find the standard deviation of the total number of pizzas produced by the shop per day.

解析

独立和的方差相加:

σT=2560=560=300.\sigma_T=\sqrt{25}\cdot 60=5\cdot 60=300.

Original Explanation

Let X1,,X25X_1, \dots, X_{25} be the number of pizzas produced by each of the workers per day. Then T=X1++X25T = X_1 + \dots + X_{25} gives the total number of pizzas produced per day by the shop. We have that Var(T)=25Var(X1)\text{Var}(T) = 25 \cdot \text{Var}(X_1) because of the fact that each of the people are independent and have the same variance. Thus, σT=Var(T)=5σX1=560=300\sigma_T = \sqrt{\text{Var}(T)} = 5 \cdot \sigma_{X_1} = 5 \cdot 60 = 300.