Probability generating functions

Probability generating functions Quiz

Spec 7.1–3 · Edexcel A-Level Further Maths: Quiz

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  1. 1. GX(t)=18(1+t)3\mathrm{G}_X(t)=\frac{1}{8}(1+t)^3. Find P(X=2)\mathrm{P}(X=2).

    • A. 18\frac18
    • B. 38\frac38
    • C. 34\frac34
    • D. 12\frac12

  2. 2. GX(t)=e3(t−1)\mathrm{G}_X(t)=\mathrm{e}^{3(t-1)}. Find E(X)\mathrm{E}(X).

    • A. 11
    • B. 3e−33\mathrm{e}^{-3}
    • C. 33
    • D. e3\mathrm{e}^3

  3. 3. GX′(1)=2\mathrm{G}_X'(1)=2 and GX′′(1)=5\mathrm{G}_X''(1)=5. Find Var(X)\mathrm{Var}(X).

    • A. 33
    • B. 11
    • C. 77
    • D. 55

  4. 4. XX and YY are independent, with GX(t)=1+t2\mathrm{G}_X(t)=\frac{1+t}{2} and GY(t)=1+t22\mathrm{G}_Y(t)=\frac{1+t^2}{2}. Find GX+Y(t)\mathrm{G}_{X+Y}(t).

    • A. 2+t+t24\frac{2+t+t^2}{4}
    • B. (1+t)(1+t2)4\frac{(1+t)(1+t^2)}{4}
    • C. 1+t2+1+t22\frac{1+t}{2}+\frac{1+t^2}{2}
    • D. 1+t32\frac{1+t^3}{2}

  5. 5. The number of trials to the second success, with success probability pp, has PGF:

    • A. 2pt1−qt\frac{2pt}{1-qt}
    • B. pt(1−qt)2\frac{pt}{(1-qt)^2}
    • C. (pt1−qt)2\left(\frac{pt}{1-qt}\right)^2
    • D. p2t1−qt\frac{p^2t}{1-qt}

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