Chapter 1 - Answers to selected exercises

1. Obtain the probability of adding up six points if we toss three distinct dice. $P=10/36.\bigskip$ 2. Consider a binomial distribution for a one-dimensional random walk, with $N=6$, $p=2/3$, and $q=1-p=1/3$. a. Draw a graph of $P_{N}(N_{1})$ versus $N_{1}/N$. b. Use the values of $\left\langle N_{1}\right\rangle$ and $\left\langle N_{1}^{2}\right\rangle$ to obtain the corresponding Gaussian distribution, $p_{G}(N_{1})$. Draw a graph of $p_{G}(N_{1})$ versus $N_{1}/N$ to compare with the previous result. c. Repeat items (a) and (b) for $N=12$ and $N=36$. Are the new answers too different?

3. Obtain an expression for the third moment of a binomial distribution. What is the behavior of this moment for large $N$? $\left\langle \left( \Delta N_{1}\right) ^{3}\right\rangle =Npq\left( q-p\right) .\bigskip$ 4. Consider an event of probability $p$. The probability of $n$ occurrences of this event out of $N$ trials is given by the binomial distribution $^{{}}$

$$W_{N}(n)=\frac{N!}{n!(N-n)!}p^{n}(1-p)^{N-n}.$$

If $p$ is small ($p<<1$), $W_{N}(n)$ is very small, except for $n<<N$. In this limit, show that we obtain the Poisson distribution,

$$W_{N}(n)\rightarrow P\left( n\right) =\frac{\lambda ^{n}}{n!}\exp \left( -\lambda \right) ,$$

where $\lambda =np$ is the mean number of events. Check that $P\left( n\right)$ is normalized. Calculate $\left\langle n\right\rangle$ and $\left\langle \left( \Delta n\right) ^{2}\right\rangle$ for this Poisson distribution. Formulate a statistical problem to be solved in terms of this distribution. Note that $\left( 1-p\right) ^{n}=\exp \left[ -pN+...\right]$, and $N!/\left( N-n\right) !=\exp \left[ n\ln N+...\right]$, for $p<<1$ and $n<<N$. Also,we have

$$\sum\limits_{n}\frac{\lambda ^{n}}{n!}=\exp \left( \lambda \right) .$$

Therefore, $\left\langle n\right\rangle =\lambda$ and $\left\langle \left( \Delta n\right) ^{2}\right\rangle =\lambda .\bigskip$ 5. Consider an experiment with $N$ equally likely outcomes, involving two events $A$ and $B$. Let $N_{1}$ be the number of events in which $A$ occurs, but not $B$; $N_{2}$ be the number of events in which $B$ occurs, but not $A$ ; $N_{3}$ be the number of events in which both $A$ and $B$ occur; and $N_{4}$ be the number of events in which neither $A$ nor $B$ occur. (a) Check that $N_{1}+N_{2}+N_{3}+N_{4}=N$. (b) Check that

$$P\left( A\right) =\frac{N_{1}+N_{3}}{N} P\left( B\right) =\frac{ N_{2}+N_{3}}{N},$$

where $P\left( A\right)$ and $P\left( B\right)$ are the probabilities of occurrence of $A$ and $B$, respectively. (c) Calculate the probability $P\left( A+B\right)$ of occurrence of either $A$ or $B$. (d) Calculate the probability $P\left( AB\right)$ of occurrence of both $A$ and $B$. (e) Calculate the conditional probability $P\left( A\mid B\right)$ that $A$ occurs given that $B$ occurs. (f) Calculate the conditional probability $P\left( B\mid A\right)$ that $B$ occurs given that $A$ occurs. (g)Show that

$$P\left( A+B\right) =P\left( A\right) +P\left( B\right) -P\left( AB\right)$$

and

$$P\left( AB\right) =P\left( B\right) P\left( A\mid B\right) =P\left( A\right) P\left( B\mid A\right) .$$

(h) Considering a third event $C$, show that

$$\frac{P\left( B\mid A\right) }{P\left( C\mid A\right) }=\frac{P\l... ...) }{P\left( C\right) }\frac{P\left( A\mid B\right) }{P\left( A\mid C\right) },$$

which is an expression of Bayes' theorem.

6. A random variable $x$ is associated with the probability density

$$p\left( x\right) =\exp \left( -x\right) ,$$

for $0<x<\infty$. (a) Find the mean value $\left\langle x\right\rangle$. (b) Two values $x_{1}$ and $x_{2}$ are chosen independently. Find $\left\langle x_{1}+x_{2}\right\rangle$ and $\left\langle x_{1}x_{2}\right\rangle$. (c) What is the probability distribution of the random variable $y=x_{1}+x_{2}$? (a) $\left\langle x\right\rangle =1$; (b) $\left\langle x_{1}+x_{2}\right\rangle =2$ and $\left\langle x_{1}x_{2}\right\rangle =1$; (c) Note that

$$p\left( y\right) =\int \int dx_{1}dx_{2}p\left( x_{1}\right) p\left( x_{2}\right) \delta \left( y-x_{1}-x_{2}\right) =y\exp \left( -y\right) .$$

To obtain this result, it is enough to use an ntegral representation of the $\delta$-function (see Appendix) and perform the integrations.

7. Consider a random walk in one dimension. After $N$ steps from the origin, the position is given by $x=\sum_{j=1}^{N}s_{j}$, where $\left\{ s_{j}\right\}$ is a set of independent, identically distributed, random variables, given by the probability distribution

$$w(s)=\left( 2\pi \sigma ^{2}\right) ^{-1/2}\exp \left[ -\frac{\left( s-l\right) ^{2}}{2\sigma ^{2}}\right] ,$$

where $\sigma$ and $l$ are positive constants. After $N$ steps, what is the average displacement from the origin? What is the standard deviation of the random variable $x$? In the large $N$ limit, what is the form of the Gaussian distribution associated with this problem? $\left\langle x\right\rangle =Nl$; $\left\langle \left( x-\left\langle x\right\rangle \right) ^{2}\right\rangle =n\sigma ^{2}$;

$$p_{G}\left( x\right) =\left( 2\pi N\sigma ^{2}\right) ^{-1/2}\exp \left[ - \frac{\left( x-Nl\right) ^{2}}{2N\sigma ^{2}}\right] .$$

8. Consider again problem 7, with a distribution $w\left( s\right)$ of the Lorentzian form

$$w(s)=\frac{1}{\pi }\frac{a}{s^{2}+a^{2}},$$

with $a>0$. Obtain an expression for the probability distribution associated with the random variable $x$. Is it possible to write a Gaussian approximation for large $N$? Why? Now we should be careful, as $\left\langle s\right\rangle =0$, but $\left\langle s^{2}\right\rangle$ diverges! The Lorentzian form does not obey the conditions of validity of the central limit theorem.