# 統計學研究所升學考題

Please answer the following questions.

(1).What is Chebyshev's inequality?

(2).Let be a random variable representing the price of a stock with a symmetric distribution of mean 80 and variance 16. Please use Chebyshev's inequality to estimate the maximum probability that the price will drop below 72.

(3).What is the weak law of large number?

(4).Please use Chebyshev's inequality to prove the weak law of large number

### 1 個解答

• 阿泰
Lv 6
1 0 年前
最佳解答

(1).What is Chebyshev's inequality?

Ans:

柴比雪夫不等式：

Pr(｜Xbar-mux｜xbar)>=1-1/k^2

Pr(｜Xbar-mux｜>k*sigmaxbar)<=1-1/k^2

(2)

Ans:

Pr(｜Xbar-mux｜>8)=Pr(｜Xbar-mux｜>8/4*4)<=1/k^2

k=2，因此Pr(｜Xbar-mux｜>8)<=1/2^2=0.25

故最大機率為0.25

(3).What is the weak law of large number?

Ans: 弱大數法則：limn->∞Pr(｜Xbar-mux｜ 0

(4).Please use Chebyshev's inequality to prove the weak law of large number

Pf:

令e=k*sigmaxbar=k*sigmax/根號{n}，因此k=根號{n}*e/sigmax

Pr(｜Xbar-mux｜x｜xbar)>=1-1/k^2=1-sigma2x/[n*e^2]

當n->∞(n趨近無限大)，sigma2x/[n*e^2]將趨近於0(因為n在分母)

因此，limn->∞Pr(｜Xbar-mux｜

2007-03-01 10:25:24 補充：

錯誤修正：

（１）

Pr(｜Xbar-mux｜>k*sigma_xbar)<=1-1/k^2

Pr(｜Xbar-mux｜<k*sigma_xbar)>=1/k^2

(３)

弱大數法則： lim　Pr(｜Xbar-mux｜<e)=1, e>0

n->∞

（４）最後　lim　Pr(｜Xbar-mux｜<e)=1

n->∞

2007-03-01 10:26:35 補充：

不好意思，有些符號被吃掉了，所以補充修正，請見諒！

2007-03-01 10:27:13 補充：

(4).

Pf:

令e=k*sigmaxbar=k*sigmax/根號{n}，因此k=根號{n}*e/sigmax

Pr(｜Xbar-mux｜<e)=Pr(｜Xbar-mux｜<k*sigmaxbar)>=1-1/k^2=1-sigma2x/[n*e^2]

當n->∞(n趨近無限大)，sigma2x/[n*e^2]將趨近於0(因為n在分母)

因此，limn->∞Pr(｜Xbar-mux｜<e)＝１

參考資料： 還蠻懂統計的我