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2014 Paper 3 Q13
D: 1700.0 B: 1500.0
probability probability generating functions PGF conditional probability geometric distribution expectation comparing coefficients

I play a game which has repeated rounds. Before the first round, my score is \(0\). Each round can have three outcomes:

  1. my score is unchanged and the game ends;
  2. my score is unchanged and I continue to the next round;
  3. my score is increased by one and I continue to the next round.
The probabilities of these outcomes are \(a\), \(b\) and \(c\), respectively (the same in each round), where \(a+b+c=1\) and \(abc\ne0\). The random variable \(N\) represents my score at the end of a randomly chosen game. Let \(G(t)\) be the probability generating function of \(N\).
  1. Suppose in the first round, the game ends. Show that the probability generating function conditional on this happening is 1.
  2. Suppose in the first round, the game continues to the next round with no change in score. Show that the probability generating function conditional on this happening is \(G(t)\).
  3. By comparing the coefficients of \(t^n\), show that $ G(t) = a + bG(t) + ctG(t)\,. $ Deduce that, for \(n\ge0\), \[ P(N=n) = \frac{ac^n}{(1-b)^{n+1}}\,. \]
  4. Show further that, for \(n\ge0\), \[ P(N=n) = \frac{\mu^n}{(1+\mu)^{n+1}}\,, \] where \(\mu=\E(N)\).

Solution:

  1. If the game ends in the first round then the score is exactly \(0\) and the pgf is \(1\cdot x^0 = 1\)
  2. If the game moves onto the next round with no change in the first round then it's as if nothing happened, therefore the pgf is the original pgf \(G(t)\)
  3. If the game moves into the next round with the score increased by one, then the pgf is \(tG(t)\) since all the scores are increased by \(1\). Therefore \begin{align*} && G(t) &= \E[t^N] \\ &&&= \E[\E[t^N | \text{first round}]] \\ &&&= a + bG(t) + ctG(t) \\ \Rightarrow && G(t)(1-b-ct) = a \\ \Rightarrow && G(t) &= \frac{a}{(1-b)-ct} \\ &&&= \frac{a}{(1-b)} \frac{1}{1- \left(\frac{c}{1-b}\right)t} \\ &&&= \sum_{n=0}^\infty \frac{a}{1-b} \frac{c^n}{(1-b)^n} t^n\\ &&&= \sum_{n=0}^{\infty} \frac{ac^n}{(1-b)^{n+1}}t^n \end{align*} Therefore by comparing coefficients, \(\mathbb{P}(N=n) = \frac{ac^n}{(1-b)^{n+1}}\)
  4. \(\,\) \begin{align*} && \E[N] &= G'(1) \\ &&&= \frac{ac}{((1-b)-c)^2} \\ &&&= \frac{ac}{a^2} = \frac{c}{a} \\ \\ && \frac{\mu^n}{(1+\mu)^{n+1}} &= \frac{c^na^{-n}}{(a+c)^{n+1}a^{-n-1}} \\ &&&= \frac{ac^n}{(a+c)^{n+1}}\\ &&&= \frac{ac^n}{(1-b)^{n+1}}\\ &&&= \mathbb{P}(N=n) \end{align*} as required

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2010 Paper 1 Q1
D: 1484.0 B: 1516.0
simultaneous equations completing the square identity quadratic comparing coefficients algebraic manipulation

Given that \[ 5x^{2}+2y^{2}-6xy+4x-4y\equiv a\left(x-y+2\right)^{2} +b\left(cx+y\right)^{2}+d\,, \] find the values of the constants \(a\), \(b\), \(c\) and \(d\). Solve the simultaneous equations \begin{align*} 5x^{2}+2y^{2}-6xy+4x-4y&=9\,, \\ 6x^{2}+3y^{2}-8xy+8x-8y&=14\,. \end{align*}

Solution: \(a\left(x-y+2\right)^{2} +b\left(cx+y\right)^{2}+d\, \equiv (a + bc^2)x^2 + (a+b)y^2 + (-2a+2bc)xy + (4a)x+(-4ay) + 4a+d\) so we want to solve \[ \begin{cases} a + bc^2 &= 5 \\ a+b &= 2 \\ 2bc - 2a &= -6 \\ 4a &= 4 \\ -4a &= 4 \\ 4a+d &= -9 \end{cases} \Rightarrow a = 1, b = 1, c = -2, d = -13 \] Therefore we have: \((x-y+2)^2 + (2x+y)^2-13 = 0\) and our simultaneous equations will be: \[ \begin{cases} (x-y+2)^2 + (-2x+y)^2 &= 13 \\ 2(x-y+2)^2 + (-2x+y)^2 &= 22 \end{cases} \] which are simultaneous equations in \((x-y+2)^2\) and \((-2x+y)^2\) which solve to \((x-y+2)^2 = 9, (-2x+y)^2 = 4 \) so we need to solve \(4\) sets of simultaneous equations: \begin{align*} &\begin{cases} x - y + 2 &= 3 \\ -2x + y &= 2 \end{cases} &&\Rightarrow (x,y) = (-3, -4) \\ &\begin{cases} x - y + 2 &= -3 \\ -2x + y &= 2 \end{cases} &&\Rightarrow (x,y) = (3, 8) \\ &\begin{cases} x - y + 2 &= 3 \\ -2x + y &= -2 \end{cases} &&\Rightarrow (x,y) = (1, 0) \\ &\begin{cases} x - y + 2 &= -3 \\ -2x + y &= -2 \end{cases} &&\Rightarrow (x,y) = (7, 12) \\ \end{align*} So \((x,y) = (-3, -4), (3, 8), (1, 0), (7,12)\)

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2009 Paper 3 Q2
D: 1700.0 B: 1484.0
second order differential equation power series series solution recurrence relation comparing coefficients trigonometric Maclaurin series

  1. Let \(\displaystyle y= \sum_{n=0}^\infty a_n x^n\,\), where the coefficients \(a_n\) are independent of \(x\) and are such that this series and all others in this question converge. Show that \[ \displaystyle y'= \sum_{n=1}^\infty na_n x^{n-1}\,, \] and write down a similar expression for \(y''\). Write out explicitly each of the three series as far as the term containing \(a_3\).
  2. It is given that \(y\) satisfies the differential equation \[ xy''-y'+4x^3y =0\,. \] By substituting the series of part (i) into the differential equation and comparing coefficients, show that \(a_1=0\). Show that, for \(n\ge4\), \[ a_n =- \frac{4}{n(n-2)}\, a_{n-4}\,, \] and that, if \(a_0=1\) and \(a_2=0\), then \( y=\cos (x^2)\,\). Find the corresponding result when \(a_0=0\) and \(a_2=1\).

Solution:

  1. Let \(\displaystyle y= \sum_{n=0}^\infty a_n x^n\,\) then \begin{align*} y' &= \frac{\d}{\d x} \l \sum_{n=0}^\infty a_n x^n \r \\ &= \sum_{n=0}^\infty \frac{\d}{\d x} \l a_n x^n \r \\ &= \sum_{n=0}^\infty n a_n x^{n-1} \\ &= \sum_{n=1}^\infty n a_n x^{n-1} \\ \\ y'' &= \frac{\d}{\d x} \l\sum_{n=1}^\infty n a_n x^{n-1} \r \\ &= \sum_{n=1}^\infty \frac{\d}{\d x} \l n a_n x^{n-1} \r \\ &= \sum_{n=1}^\infty n(n-1) a_n x^{n-2} \\ &= \sum_{n=2}^\infty n(n-1) a_n x^{n-2} \\ \\ y &= a_0 + a_1 x+ a_2x^2 + a_3x^3 + \cdots \\ y'&= a_1 + 2a_2x+3a_3x^2 + \cdots \\ y'' &= 2a_2 + 6a_3x + \cdots \end{align*}
  2. \begin{align*} && 0 &= xy''-y'+4x^3y \\ &&&= x\sum_{n=2}^\infty n(n-1) a_n x^{n-2} - \sum_{n=1}^\infty n a_n x^{n-1} + 4x^3 \sum_{n=0}^\infty a_n x^n \\ &&&= \sum_{n=2}^\infty n(n-1) a_n x^{n-1} - \sum_{n=1}^\infty n a_n x^{n-1} + \sum_{n=0}^\infty 4a_n x^{n+3} \\ &&&= \sum_{n=2}^\infty n(n-1) a_n x^{n-1} - \sum_{n=1}^\infty n a_n x^{n-1} + \sum_{n=4}^\infty 4a_{n-4} x^{n-1} \\ &&&= \sum_{n=4}^{\infty} \l n(n-1) a_n- n a_n +4a_{n-4} \r x^{n-1} + 2a_2x + 6a_3x^2-a_1-2a_2x-3a_3x^2 \\ &&&= \sum_{n=4}^{\infty} \l n(n-2) a_n +4a_{n-4} \r x^{n-1}+ 3a_3x^2-a_1 \\ \end{align*} Therefore since all coefficients are \(0\), \(a_1 = 0\), \(a_3 = 0\) and \(\displaystyle a_n = -\frac{4}{n(n-2)}a_{n-4}\). If \(a_0 = 1, a_2 = 0\), and since \(a_1 = 0, a_3 = 0\) the only values which will take non-zero value are \(a_{4k}\). We can compute these values as: \(a_{4k} = -\frac{4}{(4k)(4k-2)} a_{4k-4} = \frac{1}{2k(2k-1)}a_{4k-r}\) so \(a_{4k} = \frac{(-1)^k}{(2k)!}\), which are precisely the coefficients in the expansion \(\cos x^2\). If \(a_0 = 0, a_2 = 1\) then since \(a_1 = 0, a_3 = 0\) the only values which take non-zero values are \(a_{4k+2}\) we can compute these values as: \(a_{4k+2} = -\frac{4}{(4k+2)(4k)}a_{4k-2} = -\frac{1}{(2k+1)2k}a_{4k-2}\) so we can see that \(a_{4k+2}= \frac{(-1)^k}{(2k+1)!}\) precisely the coefficients of \(\sin x^2\)

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