TEI 發問時間： 科學數學 · 10 年前

# 統計一維隨機變數的問題

Consider the experiment of tossing two dice. Let X denote the absolute difference of the upturned faces.

(a)Find the density function of X.

(b)Find the moment generating functions of X.

(c)Find the mean and variance of X from (b)

### 1 個解答

• 最佳解答

a) px(X) = 1/36 when X = 2 or 12

px(X) = 2/36 = 1/18 when X = 3 or 11

px(X) = 3/36 = 1/12 when X = 4 or 10

px(X) = 4/36 = 1/9 when X = 5 or 9

px(X) = 5/36 when X = 6 or 8

px(X) = 6/36 = 1/6 when X = 7

px(X) = 0 otherwise

b) Moment generating function:

Mx(t) = Σ(X = 2 → 12) etX px(X)

= [e2t + e12t + 2(e3t + e11t) + 3(e4t + e10t) + 4(e5t + e9t) + 5(e6t + e8t) + 6e7t]/36

c) E[X] = Mx'(0) and E[X2] = Mx"(0)

Mx'(t) = [2e2t + 12e12t + 2(3e3t + 11e11t) + 3(4e4t + 10e10t) + 4(5e5t + 9e9t) + 5(6e6t + 8e8t) + 42e7t]/36

Mx"(t) = [4e2t + 144e12t + 2(9e3t + 121e11t) + 3(16e4t + 100e10t) + 4(25e5t + 81e9t) + 5(36e6t + 64e8t) + 294e7t]/36

E[X] = [2 + 12 + 2(3 + 11) + 3(4 + 10) + 4(5 + 9) + 5(6 + 8) + 42]/36

= 7

E[X2] = [6 + 144 + 2(9 + 121) + 3(16 + 100) + 4(25 + 81) + 5(36 + 64) + 294]/36

= 494/9

Hence variance = E[X2] - {E[X]}2 = 53/9

參考資料： 原創答案