11th January 2016 1

PYL-202, Tutorial No. 1

1. This problem is just for revising a bit about permutations and combinations. Consider throwing of

8 true dice. Find the probability of obtaining

(a) Exactly one ace.

(b) At least one ace.

(c) Exactly two aces.

2. In the class we considered Stirling’s approximation for N! ≈√

2πN NNe−N . Prove this by using

Gaussian approximation to the Gamma Integral viz. n! =∫ ∞

0e−ttn dt.

3. N gas molecules are in a container of volume V . Let v be the volume of a small region, and n

number of molecules in this region. Calculate P(n), the distribution of n. Assume the molecules

are non-interacting. Define p, probability of having a particle inside the volume v to be p = v/V .

What happens in the limit (Thermodynamic limit) N → ∞, and p → 0 such that the density

N/V → λ, where λ is a constant?

4. A long chain of molecules (freely moving in water) is composed of N links, each of length l. Let

the direction of each link be an independent variable, distributed uniformly. Calculate < R2 >,

where R is the relative position vector of the first and the last molecules. This sometimes referred

to as Freely Jointed Chain model for polymers.

5. Let the random variable x be uniformly distributed in 0 < x < 1. Calculate the distribution of

t = τ ln(1/x).

6. The step length distribution of a bacterium called E. Coli is described by a so called negative

exponential distribution given by P(α, x) = e−alphax. Is the distribution normalized? Find the mean

and variance of this distribution.