Precalculus Study Guide
Mr. Cervone
Test 3A (12/11)
4.1 Logarithmic Functions
1. Evaluating logarithms
a. A logarithm is just the inverse of an exponential function
b. log10(x) = log(x), they are the same thing.
2. Applying Properties of Logarithms
a. log
PRECALC FINAL STUDY GUIDE
5.1 The Unit Circle
1. Equation Of the Unit Circle
a. The equation for the unit circle is: x2 + y2 = 1.
Things you will have to know how to do:
a. Prove that a point lies on the unit circle
1. For Example: Does the point, (1, .25
Precalculus Study Guide
Mr. Cervone
Test 3B (1/9)
5.1 The Unit Circle
1. Equation Of the Unit Circle
a. The equation for the unit circle is: x2 + y2 = 1.
Things you will have to know how to do:
a. Prove that a point lies on the unit circle
1. For Example:
Precalculus Study Guide
Mr. Cervone
Test 4A (2/13)
5.5 Harmonic Motion
a. Simple Harmonic Motion
i. Equations: y = a sin t
y = a cos t
 a  = amplitude
period = 2 /
frequency = / 2, or 1 / period
b. Word Problems
i. Example: A mass is suspended from a s
Precalc Study Guide
Mr. Cervone
Test 1A (10/3/14)
2.1  What is a Function?
Function  a rule that assigns to each element exactly one other element in another set.
There are many types of functions

OnetoOne Functions (Injective)
Every element of the
Precalc Study Guide
Matrices > Cramers Rule
I.
The Algebra of Matrices
A. Equality of Matrices (dimensions)
1. Given that the following matrices are equal, find the values of x and y.
2. The only way for matrices to be equal is if both matrices have equa
Precalculus Study Guide
Test 6A  Polar Equations Linear Programming
I.
Polar Equations of Conics

If e < 1 then the conic is an ellipse
If e = 1 then the conic is a parabola
If e > 1 then the conic is a hyperbola
A. General form of the Polar Equation of
Precalculus Study Guide
Mr. Cervone
Test 4B (3/6/15)
8.1 Polar Coordinates
I.
Plotting Polar Coordinates
A. The polar coordinate form is as such: (r, ).
1. r represents the distance from the pole, or origin
2. represents the angle from the polar axis
B. R
Homework for 8th week.
I will be in my office during the office hours
Ch8 13, 20, 22, 24 (noticing the < and > but no = in Eq(2) of NPL), 25, 28, 29, 30,
Additonal Problems.
1. Suppose there is one observation X from the Weibull distribution with a pdf f
Homework for 10th week.
Chapter 8. 35, 44, 49
Chapter 9. 1, 2, 17,
Additonal. Answer each question clearly !
1. (b) Redo the folllowing problem and compute P (H0 H1 ) explicitly: read my solution before doing
it
Carry out the following simulation project
Homework Week 15
1. Ch10. 40, 41, 47, 48.
Additional.
A1. Answer the following questions:
If X1 , ., Xn are i.i.d. from Cauchy, where f (x) = (1 + (x)2 )1 , then
R
odd, but xf (x)dx = 2 ln(1 + x2 )
0 = .
R
xf (x)dx = 0 as xf (x) is
(a) E(X) = ?
a.s.
(b
Homework for 5th week.
Ch7. 37, 46(a,b,d) 49, 51, 52, 53, 57, 58, 59,
Additional problem:
A.1. Discuss whether the following solutions are correct for
7.12. Compare the MLE = mincfw_X, 1/2 and the MME = X.
if = X
(1 )/n if = X
E(X )2 )
=
Sol (1). M SE() =
Homework solution.
Additional.
A6.
1. Generate n=4 observations from a discrete random variable X, with X = a, b, c w.p. 1/6, 2/6, 3/6,
and with mean and variance the same as in part A5 (you need to determine a, b, c and check
help(sample) in R).
2. Perfo
Chapter 12. Decision Theory
11.1. Introduction.
There are several definitions of the optimality for an estimator.
Large samples:
1. consistency,
2. efficiency.
Small samples:
3. unbiasedness,
4. UMVUE,
5. Bayesian,
6. admissibility,
7. Minimaxity.
The las
Final Exam for Math 447, Name:
Part A. Fill the blanks of the formulas (30 points)
1. DeMorgans laws: (A B) =
and (A B) =
2. Axioms of probability: (1) P (A)
, (2) P (S) =
(3) if Ai s are pairwise
,
then P (i Ai ) =
P
i
P (Ai ).
3. If each sample point i
10.4. Approximate CI.
For finite samples, construction of CI is based on
inverting LRT and
pivotal method.
Def.
is called a Waldtype statistic if is known and if
P (
 z/2 ) 1
(1)
A. CI based on Waldtype statistic.
CI for : z/2
due to P (cfw_ :
Homework for 6th week.
A. Ch 7 44, 46(c), 48, 60, 62, 63, 65
B. Redo 7.51(d)
C. 7.51(c) Show that the MSE of T + = maxcfw_0, T is smaller than the MSE of T .
(c) Find the correct solutions among the following approaches:
(c.1) M SE(T ) = E(T )2 )
= E(T )
Homework for 7th week.
Ch8 1, 2, 3, 5, 6, 7,
Redo D. Assume that X1 , ., X100 are i.i.d. from N (0, 1). T = X 0, Y = 1(T > 1). Check which of the
following equations are correct. If so, given the explicit expressions of the density functions
involved and
Homework for week 11.
Chapter 9. 3, 12, 13, 16, 23,
Additonal.
1. A tire company guarantees that a particular brand of tire has a mean lifetime of 42 thousand miles or
more. A consumer test agency collected 10 observations as follows: 42, 36, 46, 43, 41,
Homework Week 15
1. Ch10. 31a,b,c,
Additional.
A1. Answer the following questions:
R
If X1 , ., Xn are i.i.d. from Cauchy, where f (x) = (1 + (x)2 )1 , then xf (x)dx = 0 as xf (x) is
R
Pn
1
odd, but xf (x)dx = 2 ln(1 + x2 )
0 = . Let F (t) = n
i=1 1(Xi
Homework for 2nd week.
Ch6 2,3,5,8 (give a counterexample or let f be a parameter), 9,
Additional homework.
A1. Are statements 1 and 2 equivalent ? Yes, No.
If Y = 1, then Z = X. Thus,
1. if Y = 1, then P (Z t) = P (X t);
2. P (Z t, Y = 1) = P (X t, Y = 1
Homework for 3rd week.
Ch6:10, 15,
Ch7: 1, 6, 7, 8, 9, 11, 2 (replace the last # by 1 in the data given in 7.10.c),
Additional Homework:
A1. Let X1 , ., Xn be i.i.d. with density function
f (x; ) =
1
x 
exp(
), x, (, ), > 0,
2
where = (, ). What is the
Homework for 13th week.
Ch10. 6, 7, 9,
Additional.
A1. Ch10. 5 (There is a typo in 5b : relpace V ar(Tn ) < by V ar(Tn 1(X n  > ) < ).
Do a,b, c. Moreover, under the assumption in #10.4.c, let W = Y /X = +
X,
and
compute E(W ) and V (W ) directly. (Hint:
Homework for week 11.
Chapter 9. 3, 12, 13, 16, 23,
Additonal.
1. A tire company guarantees that a particular brand of tire has a mean lifetime of 42 thousand miles or
more. A consumer test agency collected 10 observations as follows: 42, 36, 46, 43, 41,
Chapter 8, 15, 17, 33, 41, 47
Redo Ch8.7a. Solve with explicit c.
Additonal.
1. (a) Under each of the assumptions in 8.5 and 8.7, generate 10 observations from R, and do the
tests.
(b) Redo the folllowing problem and compute P (H0 H1 ) explicitly:
Carry
Homework for 14th week.
Chapter 10. 32, 34, 36, 37, 38
Additional.
A1. 1. As in Example 10.3.4, with X M ultinomial(n, p1 , ., p5 ). Set H0 : p1 = p2 = p5 , p3 = 0.5 v.s. H1 :
H0 is not true.
a. Derive the likelihood ratio test.
b. Give an estimate of P (
Homework for 4th week.
Ch7, 10, 12, 14, 23, 24, 25, 5b (use both direct derivation and R program, show me the program (see 7.2.2).
A. Additional Homework.
A1. Let X N (0, 1), W bin(2, 0.1), Y bin(1, 0.5).
X, W and Y are independent.
n
X if Y = 1
Z=
W if Y
Chapter 11. Introduction to nonparametric analysis
Common interests of estimation are
1. Mean ,
2. SD ,
3. cdf F (t).
In this course, we make the assumption that
X1 , ., Xn are i.i.d. with cdf Fo (t; ), where Fo is known, but not ( ),
Then = () and = ().
Howmework week 9 and 10
Additonal.
1. A tire company guarantees that a particular brand of tire has a mean
lifetime of 42 thousand miles or more. A consumer test agency collected 10 observations as follows: 42, 36, 46, 43, 41, 35, 43, 45, 40, 39.
Assume t