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)

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  • 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
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