Physics 471 Problem Set 1 Fall 2003
1. (Griffiths 1.1) For the distribution of ages in the example in Section 1.3,
{14,15,16,25,22,16,24,25,16,25,24,25,22,25},
(a) Compute <j
2>and <j>
2.
(b) Determine ∆jfor each jand use Eq. (1.11), σ2=<(∆j)2>, to compute the standard
deviation.
(c) Use your results in (a) and (b) to check Eq.(1.12), σ2=<j
2>−<j>
2.
2. (Griffiths 1.3) The needle on a broken car speedometer is free to swing and bounces
perfectly off the pins at either end, so that if you give it a flick it is equally likely to come
to rest at any angle between 0 and π.
(a) What is the probability density ρ(θ)? [ρ(θ)dθ is the probability that the needle will
come to rest between θand θ+dθ]. Graph ρ(θ) as a function of θ,from−π/2to
3π/2. (Of course, part of this interval is excluded, so ρis zero there.) Make sure that
the total probability is 1.
(b) Compute <θ>,<θ
2>and σfor this distribution.
(c) Compute <sin θ>,<cos θ>,and<cos2θ>.
3. (Griffiths 1.4) We consider the same device as the previous problem, but this time we are
interested in the x-coordinate of the needle point–that is, the ‘shadow’, or ‘projection’, of
the needle on the horizontal line.
(a) What is the probability density ρ(x)? [ρ(x)dx is the probability that the projection
lies between xand x+dx.] Graph ρ(x) as a function of x,from−2rto 2r,where
ris the length of the needle. Make sure the total probability is 1. [Hint: You know
from the previous problem the probability that θis in a given range; the question is,
what interval dx corresponds to the interval dθ?]
(b) Compute <x>,<x
2>and σfor this distribution.
4. Use Excel to make a histogram of the binomial distribution b(n, N, p),
b(n, N, p)=N
npnqN−n,
as a function of nfor p=0.6, N= 20, and q=1−p. Indicate the mean value <n>=Np
and the standard deviation σ=√Npq on the histogram. What is the probability that n
lies in the interval <n>−σto <n>+σ?(Note: Excel has the binomial distribution
built in.)