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2001 Paper 2 Q14
D: 1600.0 B: 1500.0
central limit theorem probability variance optimisation Bernoulli distribution linear combination normal approximation coin tossing unbiased estimator

Two coins \(A\) and \(B\) are tossed together. \(A\) has probability \(p\) of showing a head, and \(B\) has probability \(2p\), independent of \(A\), of showing a head, where \(0 < p < \frac12\). The random variable \(X\) takes the value 1 if \(A\) shows a head and it takes the value \(0\) if \(A\) shows a tail. The random variable \(Y\) takes the value 1 if \(B\) shows a head and it takes the value \(0\) if \(B\) shows a tail. The random variable \(T\) is defined by \[ T= \lambda X + {\textstyle\frac12} (1-\lambda)Y. \] Show that \(\E(T)=p\) and find an expression for \(\var(T)\) in terms of \(p\) and \(\lambda\). Show that as \(\lambda\) varies, the minimum of \(\var(T)\) occurs when \[ \lambda =\frac{1-2p}{3-4p}\;. \] The two coins are tossed \(n\) times, where \(n>30\), and \(\overline{T}\) is the mean value of \(T\). Let \(b\) be a fixed positive number. Show that the maximum value of \(\P\big(\vert \overline{T}-p\vert < b\big)\) as \(\lambda\) varies is approximately \(2\Phi(b/s)-1\), where \(\Phi\) is the cumulative distribution function of a standard normal variate and \[ s^2= \frac{p(1-p)(1-2p)}{(3-4p)n}\;. \]

Solution: \begin{align*} && \E[T] &= \E[\lambda X + \tfrac12(1-\lambda)Y] \\ &&&= \lambda \E[X] + \tfrac12(1-\lambda) \E[Y] \\ &&&= \lambda p + \tfrac12 (1-\lambda) 2p \\ &&&= p \\ \\ && \var[T] &= \var[\lambda X + \tfrac12(1-\lambda)Y] \\ &&&= \lambda^2 \var[X] + \tfrac14(1-\lambda)^2 \var[Y] \\ &&&= \lambda^2 p(1-p) + \tfrac14(1-\lambda)^22p(1-2p) \\ &&&= p(\lambda^2 + \tfrac12(1-\lambda)^2) -p^2(\lambda^2+(1-\lambda)^2)\\ &&&= p(\tfrac32\lambda^2 - \lambda + \tfrac12) -p^2(2\lambda^2 -2\lambda + 2) \end{align*} Differentiating \(\var[T]\) with respect to \(\lambda\), and noting it is a quadratic with positive leading coefficient, we get \begin{align*} && \frac{\d \var[T]}{\d \lambda} &= p(2\lambda -(1-\lambda)) - p^2(2 \lambda -2(1-\lambda)) \\ &&&= p(3\lambda - 1)-p^2(4\lambda - 2) \\ \Rightarrow && \lambda(4p-3) &= 2p-1 \\ \Rightarrow && \lambda &= \frac{1-2p}{3-4p} \end{align*} By the central limit theorem \(\overline{T} \sim N(p, \frac{\sigma^2}{n})\) in particular, \(\mathbb{P}(|\overline{T} - p| < b) = \mathbb{P}(\left \lvert |\frac{\overline{T}-p}{\frac{\sigma}{\sqrt{n}}} \right \lvert < \frac{b}{\frac{\sigma}{\sqrt{n}}}) = \mathbb{P}(|Z| < \frac{b\sqrt{n}}{\sigma}) = 2\Phi(b/s) - 1\) where \(s = \frac{\sigma}{\sqrt{n}}\) so \begin{align*} && s^2 &= \frac1n \sigma^2 \\ &&&= \frac1n \left ( \left (\left ( \frac{1-2p}{3-4p} \right)^2 + \tfrac12 \left (1-\frac{1-2p}{3-4p} \right)^2 \right)p - \left ( \left ( \frac{1-2p}{3-4p} \right)^2 + \left (1-\frac{1-2p}{3-4p} \right)^2\right)p^2 \right) \\ &&&= \frac1n \left ( \left (\left ( \frac{1-2p}{3-4p} \right)^2 + \tfrac12 \left (\frac{2-2p}{3-4p} \right)^2 \right)p - \left ( \left ( \frac{1-2p}{3-4p} \right)^2 + \left (\frac{2-2p}{3-4p} \right)^2\right)p^2 \right) \\ &&&= \frac{p}{n(3-4p)^2} \left ( (1 -4p + 4p^2 + 2-4p+2p^2) - (1-4p+4p^2+4-8p+4p^2)p \right) \\ &&&= \frac{p}{n(3-4p)^2} \left (3-13p+18p^2-8p^3 \right) \\ &&&= \frac{p}{n(3-4p)^2} (3-4p)(1-2p)(1-p) \\ &&&= \frac{p(1-p)(1-2p)}{(3-4p)n} \end{align*}

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1991 Paper 1 Q15
D: 1516.0 B: 1484.0
central limit theorem normal approximation probability fair coin random walk mean and variance dependent random variables Bernoulli distribution

A fair coin is thrown \(n\) times. On each throw, 1 point is scored for a head and 1 point is lost for a tail. Let \(S_{n}\) be the points total for the series of \(n\) throws, i.e. \(S_{n}=X_{1}+X_{2}+\cdots+X_{n},\) where \[ X_{j}=\begin{cases} 1 & \text{ if the }j \text{ th throw is a head}\\ -1 & \text{ if the }j\text{ th throw is a tail.} \end{cases} \]

  1. If \(n=10\,000,\) find an approximate value for the probability that \(S_{n}>100.\)
  2. Find an approximate value for the least \(n\) for which \(\mathrm{P}(S_{n}>0.01n)<0,01.\)
Suppose that instead no points are scored for the first throw, but that on each successive throw, 2 points are scored if both it and the first throw are heads, two points are deducted if both are tails, and no points are scored or lost if the throws differ. Let \(Y_{k}\) be the score on the \(k\)th throw, where \(2\leqslant k\leqslant n.\) Show that \(Y_{k}=X_{1}+X_{k}.\) Calculate the mean and variance of each \(Y_{k}\) and determine whether it is true that \[ \mathrm{P}(Y_{2}+Y_{3}+\cdots+Y_{n}>0.01(n-1))\rightarrow0\quad\mbox{ as }n\rightarrow\infty. \]

Solution: Notice that \(\mathbb{E}(X_i) = 0, \mathbb{E}(X_i^2) = 1\) and so \(\mathbb{E}(S_n) =0, \textrm{Var}(S_n) = n\).

  1. Then by the central limit theorem (or alternatively the normal approximation to the binomial), \begin{align*} && \mathbb{P}(S_n > 100) &\underbrace{\approx}_{\text{CLT}} \mathbb{P} \left (Z > \frac{100}{\sqrt{10\, 000}} \right) \\ &&&= \mathbb{P}(Z > 1) \\ &&&= 1-\Phi(1) \\ &&&\approx 15.9\% \end{align*}
  2. \begin{align*} &&\mathbb{P}(S_n > 0.01n) &\approx \mathbb{P} \left (Z > \frac{0.01n}{\sqrt{n}} \right) \\ &&&= \mathbb{P}(Z > 0.01 \sqrt{n}) \\ &&&= 1-\Phi(0.01\sqrt{n}) \\ &&&< 0.01 \\ && \Phi^{-1}(0.01) &= -2.3263\ldots \\ \Rightarrow && 0.01 \sqrt{n} &= 2.3263\ldots \\ \Rightarrow && n &\approx 233^2 \end{align*}
\begin{array}{cc|cc} 1\text{st throw}& k\text{th throw} & X_1 + X_k & Y_k \\ \hline \text{head} & \text{head} & 1 + 1 & 2 \\ \text{head} & \text{tail} & 1 - 1 & 0 \\ \text{tail} & \text{head} & -1 + 1 & 0 \\ \text{tail} & \text{tail} & -1- 1 & -2 \\ \end{array} Across all possible cases \(Y_k = X_1 + X_k\) so therefore these random variables are equal. \begin{align*} \mathbb{E}(Y_k) &= \mathbb{E}(X_1) + \mathbb{E}(Y_k) \\ &= 0 + 0 = 0 \\ \\ \textrm{Var}(Y_k) &= \textrm{Var}(X_1)+\textrm{Var}(X_k) \\ &= 2 \\ \\ \mathbb{E}\left (\sum_{k=2}^n Y_k \right) &= 0 \\ \textrm{Var}\left (\sum_{k=2}^n Y_k \right) &= 2(n-1) \end{align*} Therefore approximately \(\displaystyle \sum_{k=2}^n Y_k \approx N(0, 2(n-1))\) \begin{align*} \mathbb{P} \left (\sum_{k=2}^n Y_k > 0.01(n-1) \right) &\approx \mathbb{P} \left (Z > \frac{0.01(n-1)}{\sqrt{2(n-1)}} \right) \\ &= \mathbb{P} \left (Z > c \sqrt{n-1} \right) \\ &\to 0 \text{ as } n \to \infty \end{align*}

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