3
$\begingroup$

So I have this problem to solve...


Let X denote the number of paint defects found in a square yard section of a car body painted by a robot.

These data are obtained: 8, 5, 0, 10, 0, 3, 1, 12, 2, 7, 9, 6

Assume that X has a Poisson distribution with parameter λs.

a) Find an unbiased estimate for λs.

b) Find an unbiased estimate for the average number of flaws per square yard.

c) Find an unbiased estimate for the average number of flaws per square foot.


I have no idea where to begin. I mean, how do I even find ANY unbiased estimate? The textbook is worthless imo and I can't find any good readings on the web either... please help.

  • 3
    The Wikipedia page http://en.wikipedia.org/wiki/Poisson_distribution#Parameter_estimation has a simple formula for an unbiased estimation of $\lambda$. It is even the one you might guess.2010-12-07
  • 0
    @Ross no, that's not it. I need an unbiased estimate, not maximum likelihood. 2 separate things (according to my textbook at least).2010-12-07
  • 0
    It claims this is an unbiased estimator, with a short justification.2010-12-07
  • 3
    For any distribution with mean $\theta$ ($\theta = \lambda s$ in your example), the sample average is an unbiased estimator for $\theta$: $E(\bar X_n) = \theta$.2010-12-07
  • 2
    Also, what's the difference between a) and b)? Maybe "Assume that X has a Poisson distribution with parameter $\lambda$" (rather than $\lambda s$)?2010-12-07
  • 0
    This seems a rather poor simulation (low likelihood) of a Poisson distribution. I would expect some peak.2010-12-07
  • 0
    Thanks Shai, that seems reasonable and somewhat familiar. How should I accept the answer when you've put it in the comments?2010-12-07
  • 0
    I'll give some general answer later on.2010-12-07

1 Answers 1

3

In a somewhat more general setting, let $A(R)$ denote the area of region $R$. If the number of flaws found on region $R$ follows a Poisson distribution, then the mean is proportional to $A(R)$. That is, if $X(R)$ denotes the number of flaws found on region $R$, then $X(R)$ is Poisson distributed with mean $\lambda A(R)$, for some fixed $\lambda > 0$. Now, if $X_1,\ldots,X_n$ are i.i.d. Poisson$(\lambda)$ rv's (corresponding to the number of flaws found on a unit-area region, say square yard), then $\frac{1}{n}\sum\nolimits_{i = 1}^n {X_i }$ is an unbiased estimator for $\lambda$, and in turn, $\frac{1}{n}\sum\nolimits_{i = 1}^n {A(R) X_i }$ is an unbiased estimator for $\lambda A(R) = {\rm E}[X(R)]$. So, in order to estimate ${\rm E}[X(R)]$, it suffices to estimate $\lambda$ (and then just multiply by $A(R)$).