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Tabulated values of \({\Phi}(\cdot)\), the cumulative distribution function of a standard normal variable, should not be used in this question. Henry the commuter lives in Cambridge and his working day starts at his office in London at 0900. He catches the 0715 train to King's Cross with probability \(p\), or the 0720 to Liverpool Street with probability \(1-p\). Measured in minutes, journey times for the first train are \(N(55,25)\) and for the second are \(N(65,16)\). Journey times from King's Cross and Liverpool Street to his office are \(N(30,144)\) and \(N(25,9)\), respectively. Show that Henry is more likely to be late for work if he catches the first train. Henry makes \(M\) journeys, where \(M\) is large. Writing \(A\) for \(1-{\Phi}(20/13)\) and \(B\) for \(1-{\Phi}(2)\), find, in terms of \(A\), \(B\), \(M\) and \(p\), the expected number, \(L\), of times that Henry will be late and show that for all possible values of \(p\), $$BM \le L \le AM.$$ Henry noted that in 3/5 of the occasions when he was late, he had caught the King's Cross train. Obtain an estimate of \(p\) in terms of \(A\) and \(B\). [A random variable is said to be \(N\left({{\mu}, {\sigma}^2}\right)\) if it has a normal distribution with mean \({\mu}\) and variance \({\sigma}^2\).]
Solution: If Henry catches the first train, his journey time is \(N(55+30,25+144) = N(85,13^2)\). He is on time if the journey takes less than \(105\) minutes, \(\frac{20}{13}\) std above the mean. If he catches the second train, his journey times is \(N(65+25, 16+9) = N(90, 5^2)\). He is on time if his journey takes less than \(80\) minutes, ie \(\frac{10}{5} = 2\) standard deviations above the mean. This is more likely than from the first train. \(A = 1 - \Phi(20/13)\) is the probability he is late from the first train. \(B = 1 - \Phi(2)\) is the probability he is late from the second train. The expected number of lates is \(L = M \cdot p \cdot A + M \cdot (1-p) \cdot B\), since \(B \leq A\) we must have \(BM \leq L \leq AM\) \begin{align*} && \frac35 &= \frac{pA}{pA + (1-p)B} \\ \Rightarrow && 3(1-p)B &= 2pA \\ \Rightarrow && p(2A+3B) &= 3B \\ \Rightarrow && p &= \frac{3B}{2A+3B} \end{align*}