IE5002 HW1
(Deadline: 8 Feb 23:59)
1.
An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the
parameter. For example, the sample standard deviation
where
is unbiased since
is the population variance. Notice that sample standard deviation
is NOT an unbiased estimator. If
then what is
2.
? Show your calculation in detail.(Hint: Consider the chi-squared distribution.)
The failures of a machine, occurring according to a Poisson Process with rate , are registered by a counter. However,
each time an event is registered the counter becomes inoperative for the next
that might occur during that interval. Let
3.
denote the number of events that occur by time and are registered.
(a) Find the probability that the first
events are all registered.
(b) For
.
, find
The compressive strength of samples of cement can be modeled by a normal distribution with a mean of
kilograms per square centimeter and a standard deviation of
kilograms per square centimeter.
(a) What is the probability that a sample's strength is less than
(b) What is the probability that a sample's strength is between
(c) What strength is exceeded by
4.
units of time and does not register any new events
?
and
?
of the samples?
A random sample of
reduce air pollution. If more than
these fuels.
registered voters in Phoenix is asked if they favor the use of oxygenated fuels year‐round to
voters respond positively, we will conclude that at least
(a) Find the probability of a type I error if exactly
(b) What is the type II error probability
if
of the voters favor the use of these fuels.
of the voters favor this action?
of the voters favor the use of