Atmospheric dust particles around construction sites cause environmental problems. The number of particles in a unit volume is modeled as a Poisson distribution with unknown parameter lambda. 50 small air samples were collected containing a total of 2500 particles. The maximum likelihood estimate of lambda is calculated as 2500/50 = 50. In traffic engineering, the time intervals between vehicles passing a point are exponentially distributed with unknown parameter lambda. For 30 observed intervals with a mean of 0.55 minutes, the maximum likelihood estimate of lambda is calculated.

1. Please provide solutions:
Atmospheric dust particles around construction sites cause a serious environmental problem. It
is assumed that the number of particles in a unit volume is Poisson whose parameter lambda
needs to be estimated. In order to do it a small randomly selected portion of the air was observed
by focusing a powerful microscope on the particles and making counts. Suppose 50 such samples
of air were collected and it gave a total of 2500 particles. Find the m.l.e. of lambda . Let where
X is an exponential r.v. with parameter lambda unknown. Find the m.l.e. and MLE of lambda . Is
this MLE unbiased? In traffic engineering one is concerned with the length of time lambda
between vehicles passing a given point. If the intervals are too short, there will be halts and
interruptions, which the engineers attempt to minimize by imposing longer waiting times at the
associated lights. The time intervals between vehicles passing a point of observation are known
to be exponentially distributed with an unknown parameter lambda . In the context of Problem F.
suppose there are 30 observed intervals between the passings of vehicular traffic giving the mean
time interval of 0.55 minute. Find the m.l.e. of lambda .
Solution
E. mle = 2500/50= 50
F. if X= exponetential r.v them mle = zero
G. mle= 0 if parameter is exponential
may refer to following paper if
neededhttp://www.math.ucsd.edu/~gptesler/283/slides/283_mle_11-handout.pdf
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