Exercises

1. Let $I_x= \int_{\-\infty}^\infty {\rm e}^{-x^{ 2}} dx$ and $I_y= \int_{\-\infty}^\infty {\rm e}^{-y^{ 2}} dy$ . Show that
   $$I_x I_y= \int_0^\infty 2\pi r {\rm e}^{-r^{ 2}} dr,$$
   where $r^{ 2}=x^{ 2}+y^{ 2}$ . Hence, deduce that $I_x=I_y=\pi^{ 1/2}$ .

2. Show that
   $$\int_{-\infty}^{\infty} {\rm e}^{-\beta x^{ 2}} dx = \sqrt{\frac{\pi}{\beta}}.$$
   Hence, deduce that
   $$\int_{-\infty}^\infty x^{ 2} {\rm e}^{-\beta x^{ 2}} dx=\frac{\pi^{ 1/2}}{2 \beta^{ 3/2}}.$$

3. Confirm that

   $$\int_{-\infty}^\infty {\cal P}(n) dn =1,$$

   $$\int_{-\infty}^\infty {\cal P}(n) n dn =\bar{n},$$

   $$\int_{-\infty}^\infty {\cal P}(n) (n-\bar{n})^{ 2} dn ={\mit\Delta}^\ast n,$$

   
   
   where
   $${\cal P}(n)=\frac{1}{\sqrt{2\pi} {\mit\Delta}^\ast n}\exp\left[-\frac{(n-\bar{n})^{ 2}}{2 ({\mit\Delta}^\ast n)^{ 2}}\right].$$

4. Show that the probability of throwing 6 points or less with three (six-sided) dice is $5/54$ .

5. Consider a game in which six (six-sided) dice are rolled. Find the probability of obtaining:
   1. exactly one ace.
   2. at least one ace.
   3. exactly two aces.

6. In the game of Russian roulette, one inserts a single cartridge into the drum of a revolver, leaving the other five chambers of the drum empty. One then spins the drum, aims at one's head, and pulls the trigger.
   1. Show that the probability of still being alive after playing the game $N$ times is $(5/6)^{ N}$ .
   2. Show that the probability of surviving $(N-1)$ turns in this game, and then being shot the $N$ th times one pulls the trigger, is $(5/6)^{ N} (1/6)$ .
   3. Show that the mean number of times a player gets to pull the trigger is $6$ .

7. A battery of total emf $V$ is connected to a resistor $R$ . As a result, an amount of power $P=V^{ 2}/R$ is dissipated in the resistor. The battery itself consists of $N$ individual cells connected in series, so that $V$ is equal to the sum of the emf's of all these cells. The battery is old, however, so that not all cells are in perfect condition. Thus, there is a probability $p$ that the emf of any individual cell has its normal value $v$ ; and a probability $1-p$ that the emf of any individual cell is zero because the cell has become internally shorted. The individual cells are statistically independent of each other. Under these conditions, show that the mean power, $\overline{P}$ , dissipated in the resistor, is
   $$\overline{P}= \frac{p^{ 2} V^{ 2}}{R}\left[1-\frac{(1-p)}{N p}\right].$$

8. A drunk starts out from a lamppost in the middle of a street, taking steps of uniform length $l$ to the right or to the left with equal probability.
   1. Show that the average distance from the lamppost after $N$ steps is zero.
   2. Show that the root-mean-square distance (i.e. the square-root of the mean of the distance squared) from the lamppost after $N$ steps is $\sqrt{N} l$ .
   3. Show that the probability that the drunk will return to the lamppost after $N$ steps is zero if $N$ is odd, and
      $$P_N = \frac{N!}{(N/2)! (N/2)!}\left(\frac{1}{2}\right)^N$$
      if $N$ is even.

9. A molecule in a gas moves equal distances $l$ between collisions with equal probabilities in any direction. Show that, after a total of $N$ such displacements, the mean-square displacement, $\overline{R^{ 2}}$ , of the molecule from its starting point is $\overline{R^{ 2}}=N l^{ 2}$ .

10. A penny is tossed 400 times. Find the probability of getting 215 heads. (Use the Gaussian distribution.)

11. Suppose that the probability density for the speed $s$ of a car on a highway is given by
    $$\rho(s) = A s \exp\left(\frac{-s}{s_0}\right),$$
    where $0\leq s\leq \infty$ . Here, $A$ and $s_0$ are positive constants. More explicitly, $\rho(s) ds$ gives the probability that a car has a speed between $s$ and $s+ds$ .
    1. Determine $A$ in terms of $s_0$ .
    2. What is the mean value of the speed?
    3. What is the ``most probable'' speed: that is, the speed for which the probability density has a maximum.
    4. What is the probability that a car has a speed more than three times as large as the mean value?