hce_nthu
111年
資訊科學
第 24 題
There are 2 exams in Statistics class. The midterm score (X) is approximately normal distributed with mean 60 and standard deviation 20. The final exam score (Y) is approximately normal distributed with mean 80 and standard deviation 10. The overall score is the weighted average of two scores: Z = X * 40% + Y * 60%.
Suppose that $\text{cor}(X,Y) = 0.6$. Which of the following statement is FALSE?
Suppose that $\text{cor}(X,Y) = 0.6$. Which of the following statement is FALSE?
- A The overall score has a mean 72
- B The overall score has a standard deviation 14
- C Emily got 60 in the midterm. The expected score for her final exam is 80
- D $\text{cor}(Z,X) > 0.6$
- E The overall score is approximately normally distributed
思路引導 VIP
若我們將兩個「正相關」的變數進行加權相加，你認為這兩個變數的波動會因為彼此的關聯而互相抵銷，還是會因為朝著相似方向變動而讓整體的波動（變異程度）變得更大呢？這對我們計算標準差時的最後一項「協方差」項會有什麼影響？
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AI 詳解
AI 專屬家教
恭喜你準確地抓出了選項 (B) 這個錯誤描述！這代表你對於隨機變數的線性組合（Linear Combination）有著非常紮實的理解。這道題目難度設定在 Medium（中等），它不只需要你記憶公式，更考驗你是否能靈活運用協方差（Covariance）來處理兩個非獨立變數之間的互動關係。
加權總分的變異數分析
在計算總分 $Z = 0.4X + 0.6Y$ 的標準差時，最關鍵的陷阱在於變數 $X$ 與 $Y$ 並非獨立，而是存在正相關（$\text{cor}(X,Y) = 0.6$）。根據變異數公式：
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