Problems

Let \(\F(x)\) be the cumulative distribution function of a random variable \(X\), which satisfies \(\F(a)=0\) and \(\F(b)=1\), where \(a>0\). Let \[ \G(y) = \frac{\F(y)}{2-\F(y)}\;. \] Show that \(\G(a)=0\,\), \(\G(b)=1\,\) and that \(\G'(y)\ge0\,\). Show also that \[ \frac12 \le \frac2{(2-\F(y))^2} \le 2\;. \] The random variable \(Y\) has cumulative distribution function \(\G(y)\,\). Show that \[ { \tfrac12} \,\E(X) \le \E(Y) \le 2 \E(X) \;, \] and that \[ \var(Y) \le 2\var(X) +\tfrac 74 \big(\E(X)\big)^2\;. \]

A densely populated circular island is divided into \(N\) concentric regions \(R_1\), \(R_2\), \(\ldots\,\), \(R_N\), such that the inner and outer radii of \(R_n\) are \(n-1\) km and \(n\) km, respectively. The average number of road accidents that occur in any one day in \(R_n\) is \(2-n/N\,\), independently of the number of accidents in any other region. Each day an observer selects a region at random, with a probability that is proportional to the area of the region, and records the number of road accidents, \(X\), that occur in it. Show that, in the long term, the average number of recorded accidents per day will be \[ 2-\frac16\left(1+\frac1N\right)\left(4-\frac1N\right)\;. \] [Note: \(\sum\limits_{n=1}^N n^2 = \frac16 N(N+1)(2N+1) \;\).] Show also that \[ \P(X=k) = \frac{\e^{-2}N^{-k-2}}{k!}\sum_{n=1}^N (2n-1)(2N-n)^k\e^{n/N} \;. \] Suppose now that \(N=3\) and that, on a particular day, two accidents were recorded. Show that the probability that \(R_2\) had been selected is \[ \frac{48}{48 + 45\e^{1/3} +25 \e^{-1/3}}\;. \]

In a game, a player tosses a biased coin repeatedly until two successive tails occur, when the game terminates. For each head which occurs the player wins \(\pounds 1\). If \(E\) is the expected number of tosses of the coin in the course of a game, and \(p\) is the probability of a head, explain why \[ E = p \l 1 + E \r + \l 1 - p \r p \l 2 + E \r + 2 \l 1 - p \r ^2\,, \] and hence determine \(E\) in terms of \(p\). Find also, in terms of \(p\), the expected winnings in the course of a game. A second game is played, with the same rules, except that the player continues to toss the coin until \(r\) successive tails occur. Show that the expected number of tosses in the course of a game is given by the expression \(\displaystyle {1 - q^r \over p q^r}\,\), where \(q = 1 - p\).

A continuous random variable is said to have an exponential distribution with parameter \(\lambda\) if its density function is \(\f(t) = \lambda \e ^{- \lambda t} \; \l 0 \le t < \infty \r\,\). If \(X_1\) and \(X_2\), which are independent random variables, have exponential distributions with parameters \(\lambda_1\) and \(\lambda_2\) respectively, find an expression for the probability that either \(X_1\) or \(X_2\) (or both) is less than \(x\). Prove that if \(X\) is the random variable whose value is the lesser of the values of \(X_1\) and \(X_2\), then \(X\) also has an exponential distribution. Route A and Route B buses run from my house to my college. The time between buses on each route has an exponential distribution and the mean time between buses is 15 minutes for Route A and 30 minutes for Route B. The timings of the buses on the two routes are independent. If I emerge from my house one day to see a Route A bus and a Route B bus just leaving the stop, show that the median wait for the next bus to my college will be approximately 7 minutes.

Prove that, for any two discrete random variables \(X\) and \(Y\), \[ \mathrm{Var} \left(X + Y \right) = \mathrm{Var}(X) + \mathrm{Var}(Y) + 2 \, \mathrm{Cov}(X,Y), \] where \(\mathrm{Var}(X)\) is the variance of \(X\) and \(\mathrm{Cov}(X,Y)\) is the covariance of \(X\) and \(Y\). When a Grandmaster plays a sequence of \(m\) games of chess, she is, independently, equally likely to win, lose or draw each game. If the values of the random variables \(W\), \(L\) and \(D\) are the numbers of her wins, losses and draws respectively, justify briefly the following claims:

1. \(W + L + D\) has variance \(0\,\);
2. \(W + L\) has a binomial distribution.

Four students, Arthur, Bertha, Chandra and Delilah, exchange gossip. When Arthur hears a rumour, he tells it to one of the other three without saying who told it to him. He decides whom to tell by choosing at random amongst the other three, omitting the ones that he knows have already heard the rumour. When Bertha, Chandra or Delilah hear a rumour, they behave in exactly the same way (even if they have already heard it themselves). The rumour stops being passed round when it is heard by a student who knows that the other three have already heard it. Arthur starts a rumour and tells it to Chandra. By means of a tree diagram, or otherwise, show that the probability that Arthur rehears it is \(3/4\). Find also the probability that Bertha hears it twice and the probability that Chandra hears it twice.

Show Solution

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Solution: Without loss of generality, \(C\) will tell \(B\) about the rumour. If \(B\) tells \(D\) then \(D\) can either tell \(A\) or \(C\) at which point either \(A\) is told or the rumour stops spreading.

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Therefore \(\mathbb{P}(\text{Arthur rehears}) = 3/4\) For the chances Chandra hears it twice, still WLOG, assume she tells B:

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So her chance of hearing it twice is \(\frac12\) The person who hears it 3rd has a \(\frac12\) chance of hearing it twice, but the person who hears if 4th has no chance. Therefore Bertha has a \(\frac14\) chance of hearing it twice.

Four students, one of whom is a mathematician, take turns at washing up over a long period of time. The number of plates broken by any student in this time obeys a Poisson distribution, the probability of any given student breaking \(n\) plates being \(\e^{-\lambda} \lambda^n/n!\) for some fixed constant \(\lambda\), independent of the number of breakages by other students. Given that five plates are broken, find the probability that three or more were broken by the mathematician.

On the basis of an interview, the \(N\) candidates for admission to a college are ranked in order according to their mathematical potential. The candidates are interviewed in random order (that is, each possible order is equally likely).

1. Find the probability that the best amongst the first \(n\) candidates interviewed is the best overall.
2. Find the probability that the best amongst the first \(n\) candidates interviewed is the best or second best overall.

The national lottery of Ruritania is based on the positive integers from \(1\) to \(N\), where \(N\) is very large and fixed. Tickets cost \(\pounds1\) each. For each ticket purchased, the punter (i.e. the purchaser) chooses a number from \(1\) to \(N\). The winning number is chosen at random, and the jackpot is shared equally amongst those punters who chose the winning number. A syndicate decides to buy \(N\) tickets, choosing every number once to be sure of winning a share of the jackpot. The total number of tickets purchased in this draw is \(3.8N\) and the jackpot is \(\pounds W\). Assuming that the non-syndicate punters choose their numbers independently and at random, find the most probable number of winning tickets and show that the expected net loss of the syndicate is approximately \[ N\; - \; %\textstyle{ \frac{5 \big(1- e^{-2.8}\big)}{14} \;W\;. \]

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 \[ s^2= \frac{p(1-p)(1-2p)}{(3-4p)n}\;. \]

A bag contains \(b\) black balls and \(w\) white balls. Balls are drawn at random from the bag and when a white ball is drawn it is put aside.

1. If the black balls drawn are also put aside, find an expression for the expected number of black balls that have been drawn when the last white ball is removed.
2. If instead the black balls drawn are put back into the bag, prove that the expected number of times a black ball has been drawn when the first white ball is removed is \(b/w\,\). Hence write down, in the form of a sum, an expression for the expected number of times a black ball has been drawn when the last white ball is removed.

In a game for two players, a fair coin is tossed repeatedly. Each player is assigned a sequence of heads and tails and the player whose sequence appears first wins. Four players, \(A\), \(B\), \(C\) and \(D\) take turns to play the game. Each time they play, \(A\) is assigned the sequence TTH (i.e.~Tail then Tail then Head), \(B\) is assigned THH, \(C\) is assigned HHT and \(D\) is assigned~HTT.

1. \(A\) and \(B\) play the game. Let \(p_{\mathstrut\mbox{\tiny HH}}\), \(p_{\mathstrut\mbox{\tiny HT}}\), \(p_{\mathstrut\mbox{\tiny TH}}\) and \(p_{\mathstrut\mbox{\tiny TT}}\) be the probabilities of \(A\) winning the game given that the first two tosses of the coin show HH, HT, TH and TT, respectively. Explain why \(p_{\mathstrut\mbox{\tiny TT}} = 1\,\), and why $p_{\mathstrut\mbox{\tiny HT}} = {1 \over 2} \, p_{\mathstrut\mbox{\tiny TH}} + {1\over 2} \, p_{\mathstrut\mbox{\tiny TT}}\,$. Show that $p_{\mathstrut\mbox{\tiny HH}} = p_{\mathstrut\mbox{\tiny HT}} = {2 \over 3}$ and that \(p_{\mathstrut\mbox{\tiny TH}} = {1\over 3}\,\). Deduce that the probability that A wins the game is \({2\over 3}\,\).
2. \(B\) and \(C\) play the game. Find the probability that \(B\) wins.
3. Show that if \(C\) plays \(D\), then \(C\) is more likely to win than \(D\), but that if \(D\) plays \(A\), then \(D\) is more likely to win than \(A\).

I have \(k\) different keys on my key ring. When I come home at night I try one key after another until I find the key that fits my front door. What is the probability that I find the correct key in exactly \(n\) attempts in each of the following three cases?

1. At each attempt, I choose a key that I have not tried before but otherwise each choice is equally likely.
2. At each attempt, I choose a key from all my keys and each of the \(k\) choices is equally likely.
3. At the first attempt, I choose from all my keys and each of the \(k\) choices is equally likely. Thereafter, I choose from the keys that I did not try the previous time but otherwise each choice is equally likely.

Every person carries two genes which can each be either of type \(A\) or of type \(B\). It is known that \(81\%\) of the population are \(AA\) (i.e. both genes are of type \(A\)), \(18\%\) are \(AB\) (i.e. there is one gene of type \(A\) and one of type \(B\)) and \(1\%\) are \(BB\). A child inherits one gene from each of its parents. If one parent is \(AA\), the child inherits a gene of type \(A\) from that parent; if the parent is \(BB\), the child inherits a gene of type \(B\) from that parent; if the parent is \(AB\), the inherited gene is equally likely to be \(A\) or \(B\).

1. Given that two \(AB\) parents have four children, show that the probability that two of them are \(AA\) and two of them are \(BB\) is \(3/128\).
2. My mother is \(AB\) and I am \(AA\). Find the probability that my father is \(AB\).

The random variable \(X\) is uniformly distributed on the interval \([-1,1]\). Find \(\E(X^2)\) and \(\var (X^2)\). A second random variable \(Y\), independent of \(X\), is also uniformly distributed on \([-1,1]\), and \(Z=Y-X\). Find \(\E(Z^2)\) and show that \(\var (Z^2) = 7 \var (X^2)\).