SECTION 6.3 Parameters and Statistics 373
(b) Conclude thatf∗f is not
differentiable atx= 0.
(c) Show that the graph of
f∗f is as depicted to the
right.(d) Also, compute∫∞
−∞f∗f(x)dx, and compare with∫∞
−∞f(x)dx.
Does this make sense? Can you formulate a general statement?6.3.2 Statistics: sample mean and variance
In all of the above discussions, we have either been dealing with ran-
dom variables whose distributions are known, and hence its mean and
variance can (in principle) be computed, or we have been deriving the-
oretical aspects of the mean and variance of a random variable. While
interesting and important, these are intellectual luxuries that usually
don’t present themselves in the real world. If, for example, I was
charged with the analysis of the mean number of on-the-job injuries
in a company in a given year, I would be tempted to model this with
a Poisson distribution. Even if this were a good assumption, I proba-
bly wouldn’t know the mean of this distribution. Arriving at a “good”
estimate of the mean and determining whether the Poisson model is a
“good” model are both statistical questions.
Estimation of a random variable’s mean will be the main focus of the
remainder of the present chapter, with a final section on the “goodness
of fit” of a model.
We turn now tostatistics. First of all, any particular values (out-
comes) of a random variable or random variables are collectively known
asdata. A statistic is any function of the data. Two particularly
important statistics are as follows. Asample(of sizen) from a distri-
bution with random variableXis a set ofnindependent measurements