Physics 4311: Thermal Physics – Homework 1

Problem 1: Joint probabilities (13 points)

The random variables $x$ and $y$ are jointly distributed. $x$ can take values 1, 2, or 3, whereas $y$ can take the values 7 or 8. The joint probabilities are given by $p_{\mathit{xy}}(1,7)=3∕8$, $p_{\mathit{xy}}(2,7)=1∕8$, $p_{\mathit{xy}}(3,7)=1∕4$, $p_{\mathit{xy}}(1,8)=1∕8$, $p_{\mathit{xy}}(2,8)=1∕24$, $p_{\mathit{xy}}(3,8)=1∕12$.

a)

Check that $p_{\mathit{xy}}$ is properly normalized.

b)

Compute the reduced probabilities $p_x(1)$, $p_x(2)$, and $p_x(3)$.

c)

Compute the reduced probabilities $p_y(7)$, and $p_y(8)$

d)

Compute the conditional probabilities $p_x(2|y=7)$ and $p_x(2|y=8)$.

e)

Determine whether or not $x$ and $y$ are statistically independent.

Problem 2: Gaussian distribution (15 points)

The continuous random variable $x$ has the probability density

for all real $x$ (where $x_0$, $A$, and $C$ are constants).

a)

Find the value of the constant $C$ (in terms of $A$ and $x_0$) such that the probability density is properly normalized.

b)

Compute the average $⟨x⟩$, the median $x_M$ and the most probable value $x_P$.

c)

Compute the second moment $⟨x^2⟩$ and the variance ${\sigma }_x^2$.

Problem 3: Probability of a 10% density fluctuation (12 points)

Consider two identical boxes, A and B.

a)

20 particles are distributed over the two identical boxes A and B at random. Calculate the probabilities $P(9)$ and $P(10)$ for finding exactly $N_A=10$ and $N_A=11$ particles in box A, respectively. Calculate the ratio $P(10)∕P(11)$.

b)

Repeat the calculations for $200$ particles. Compare the probabilities for $N_A=100$ and $N_A=110$.

c)

Repeat the calculations for $2000$ particles. Compare the probabilities for $N_A=1000$ and $N_A=1100$.

(Hint: If your calculator cannot handle large factorials, you can either use Stirling’s approximation formula $n!\approx \sqrt{2\mathit{\pi n}} n^n{e}^{-n}$ or math software such as Wolfram Alpha.)