Problems

A comet in deep space picks up mass as it travels through a large stationary dust cloud. It is subject to a gravitational force of magnitude \(M\!f\) acting in the direction of its motion. When it entered the cloud, the comet had mass \(M\) and speed \(V\). After a time \(t\), it has travelled a distance \(x\) through the cloud, its mass is \(M(1+bx)\), where~\(b\) is a positive constant, and its speed is \(v\).

1. In the case when \(f=0\), write down an equation relating \(V\), \(x\), \(v\) and \(b\). Hence find an expression for \(x\) in terms of \(b\), \(V\) and \(t\).
2. In the case when \(f\) is a non-zero constant, use Newton's second law in the form \[ \text{force} = \text{rate of change of momentum} \] to show that \[ v = \frac{ft+V}{1+bx}\,. \] Hence find an expression for \(x\) in terms of \(b\), \(V\), \(f\) and \(t\). Show that it is possible, if \(b\), \(V\) and \(f\) are suitably chosen, for the comet to move with constant speed. Show also that, if the comet does not move with constant speed, its speed tends to a constant as \(t\to\infty\).

1. Albert tosses a fair coin \(k\) times, where \(k\) is a given positive integer. The number of heads he gets is \(X_1\). He then tosses the coin \(X_1\) times, getting \(X_2\) heads. He then tosses the coin \(X_2\) times, getting \(X_3\) heads. The random variables \(X_4\), \(X_5\), \(\ldots\) are defined similarly. Write down \(\E(X_1)\). By considering \(\E(X_2 \; \big\vert \; X_1 = x_1)\), or otherwise, show that \(\E(X_2) = \frac14 k\). Find \(\displaystyle \sum_{i=1}^\infty \E(X_i)\).
2. Bertha has \(k\) fair coins. She tosses the first coin until she gets a tail. The number of heads she gets before the first tail is \(Y_1\). She then tosses the second coin until she gets a tail and the number of heads she gets with this coin before the first tail is \(Y_2\). The random variables \(Y_3\), \(Y_4\), \(\ldots\;\), \(Y_k\) are defined similarly, and \(Y= \sum\limits_{i=1}^k Y_i\,\). Obtain the probability generating function of \(Y\), and use it to find \(\E(Y)\), \(\var(Y)\) and \(\P(Y=r)\).

1. The point \(P\) lies on the circumference of a circle of unit radius and centre \(O\). The angle,~\(\theta\), between \(OP\) and the positive \(x\)-axis is a random variable, uniformly distributed on the interval \(0\le\theta<2\pi\). The cartesian coordinates of \(P\) with respect to \(O\) are \((X, Y)\). Find the probability density function for \(X\), and calculate \(\var (X)\). Show that \(X\) and \(Y\) are uncorrelated and discuss briefly whether they are independent.
2. The points \(P_i\) (\(i=1\), \(2\), \(\ldots\) , \(n\)) are chosen independently on the circumference of the circle, as in part (i), and have cartesian coordinates \((X_i, Y_i)\). The point \(\overline P\) has coordinates \((\overline X, \overline Y)\), where \(\overline X =\dfrac1n \sum\limits _{i=1}^n X_i\) and \(\overline Y =\dfrac1n \sum\limits _{i=1}^n Y_i\). Show that \(\overline X\) and \(\overline Y\) are uncorrelated. Show that, for large \(n\), $\displaystyle \P\left(\vert \overline X \vert \le \sqrt{\frac2n}\right) \approx 0.95\,$.