172 Some Special Distributions3.2.5.LetXhave a Poisson distribution withμ= 100. Use Chebyshev’s inequality
to determine a lower bound forP(75<X<125). Next, calculate the probability
using R. Is the approximation by Chebyshev’s inequality accurate?
3.2.6. The following R code segment computes a page of plots for Poisson pmfs
with means 2, 4 , 6 , ...,18. Run this code and comment on the the shapes and modes
of the distributions.
par(mfrow=c(3,3)); x= 0:35; lam=seq(2,18,2);
for(y in lam){plot(dpois(x,y)~x); title(paste("Mean is ",y))}
3.2.7.By Exercise 3.2.6 it seems that the Poisson pmf peaks at its meanλ. Show
that this is the case by solving the inequalities [p(x+1)/p(x)]>1and[p(x+
1)/p(x)]<1, wherep(x) is the pmf of a Poisson distribution with parameterλ.
3.2.8.Using the computer, obtain an overlay plot of the pmfs of the following two
distributions:
(a)Poisson distribution withλ=2.(b)Binomial distribution withn= 100 andp=0.02.Why would these distributions be approximately the same? Discuss.
3.2.9. Continuing with Exercise 3.2.8, make a page of four overlay plots for the
following 4 Poisson and binomial combinations:λ=2,p=0.02;λ=10,p=0.10;
λ=30,p=0.30;λ=50,p=0.50. Usen= 100 in each situation. Plot the subset of
the binomial range that is betweennp±
√
np(1−p). For each situation, comment
on the goodness of the Poisson approximation to the binomial.3.2.10.The approximation discussed in Exercise 3.2.8 can be made precise in the
following way. SupposeXnis binomial with the parametersnandp=λ/n,fora
givenλ>0. LetY be Poisson with meanλ. Show thatP(Xn=k)→P(Y =k),
asn→∞, for an arbitrary but fixed value ofk.
Hint:First show that:
P(Xn=k)=λk
kn
.3.2.11.Let the number of chocolate chips in a certain type of cookie have a Poisson
distribution. We want the probability that a cookie of this type contains at least
two chocolate chips to be greater than 0.99. Find the smallest value of the mean
that the distribution can take.
3.2.12.Compute the measures of skewness and kurtosis of the Poisson distribution
with meanμ.
3.2.13. On the average, a grocer sells three of a certain article per week. How
many of these should he have in stock so that the chance of his running out within
a week is less than 0.01? Assume a Poisson distribution.