MESF5450 Spring 2024/2025 Homework #1

Java Python MESF5450 Spring 2024/2025

Homework #1

1. The probability W(n) that an event characterized by a probability p occurs n times in N trials was shown to be given by the binomial distribution

Consider a situation where the probability p is small (p ≪ 1) and where one is interested in the case n ≪ N. (Note that if N is large, W(n) becomes very small if n → N because of the smallness of the factor pn  when p ≪ 1. Hence W(n) is indeed only appreciable when n ≪ N).  Several approximations can then be made to reduce the above equation to a simple form.

(a) Using the result ln(1 - p) ≈ -p, show that (1-p)N-n ≈ e-Np (b) Show that N!/(N - n)! ≈ Nn

(c) Hence shows that the above equation reduces to

2. A number is chosen at random between 0 and 1. What is the probability that exactly 5 of its first decimal places consist of digits less than 5?

3. Consider the random walk problem with p = q and let m = n1 – n2  denote the net displacement to the right. After a total of N steps, calculate the following mean values: m , m2  , m3   and m4  .

4. Derive the binomial distribution in the following algebraic way, which does not involve any explicit combinational analysis. One is again interested in finding the probability W(n) of n successes out of a total of N independent trials. Let w1 = p denote the probability of a success, w2 = 1 –p = q the corresponding probability of a failure. ThenW(n) can be obtained by writing

Here,  each term  contains N factors  and  is the probability  of a particular  combination  of successes and failures. The sum over all combinations is then to betaken only over those terms involving w1 exactly n times, i.e., only over those terms involving w1n .

By rearranging the sum of Eq. (1), show that the unrestricted sum can be written in the form.

Expanding this by the binomial theorem to show that the sum of all terms in Eq. (1) involving w1n, i.e., the desired probability W(n), is then simply given by the one binomial expansion term that involves w1n .

5. Two drunks start out together at the origin, each having an equal probability of making a step to the left or right along the x-axis. Find the probability that they meet again after N steps. It is to be understood that the men make their steps simultaneously