L a j P N j ai P N i
L a j P N j a1 P N 1 a1 a2 P N 2 a1 a2 a3 P N 3 L
a1 P N 1 P N 2 P N 3 L
a2 P N 2 P N 3 L L
a1 P N 1 a2 P N 2 L
ai P N i
2 L a j P N j ai P N i
Show that E
Let Z be a standard normal random variable and let g be a differentiable function with
derivative g .
Show that E g Z E Zg Z
E Zg Z
zg z f z dz g z f z dz E g Z
Show that E Z n 1 nE Z n 1
E Z n 1
c E Z
z n 1 f z dz
nz n 1 f z dz n z n
5) Explain a theoretical reason why agency size, measured as an ordinal scale of full-time
equivalent sworn staff, could be associated with the primary method for transmitting criminal
incident reports from the field to the agencys central information sys
1. a Let Z have a standard normal distribution.
2 2 dz
Let Y e Z Z ln Y
F Y P Y y P e Z y P Z ln y
ln y 2 dy
z ln y
ln y 2
0 y .
f y y 2
2. The distance from breakdown to city A has a uniform distribution over 0,200 .
There is a bus service station in city A x 0 , in city B x 200 , and in the center
of the route between A and B x 100 , i.e. located at 100 km. Then define the distance
The sufficient conditions for any estimator T to be consistent estimator for the parameter
E T and Var T 0 as n .
Any estimator T is said to be an unbiased estimator for the parameter , if E T .
Here, X i ~ , 2
i .i . d
X1 X 2 X 3
is a consistent e
1. How would you define the Wilcoxon Signed Rank Test?
Wilcoxon Signed Rank Test:
Let 'e the median of the population under study with density f x .
The null and alternative hypotheses for the Wilcoxon Signed Rank Test are,
H 0 : % %
H a : % %
1. Coffee Consumption [5 points]
Coffee is a very popular beverage among Americans. Chain restaurants that are coffee only like
Starbucks or Biggby are everywhere these days. In the United States, the average number of
cups of coffee consumed by Americans
A rare disease exists in which only 1 in 500 are affected. A test for the disease exists
but of course it is not infallible. A correct positive result (patient actually has the disease)
occurs 95% of the time while a false positive result (pati
(1) Forty-nine items are randomly selected from a population of 500 items. The sample mean is
40 and the sample standard deviation 9. Develop a 99% confidence interval for the population
mean. (Round your answers to 3 decimal places.)
Sample size n
1. Below we list several variables. Which of these variables are quantitative and which are
a. The dollar amount on an accounts receivable invoice.
b. The net profit for a company in 2010.
c. The stock exchange on which