(II) Jointly Distributed Continuous
Random Variables
For two continuous random variables
X and Y, the joint probability density
function is given by f (x, y),
where
f (x, y) 0 for all x and y and
f ( x, y) dx dy
=1
To compute probabilities, we simply
i
Jointly Distributed Random Variables
(section 2.5)
So far we have studied probability
distributions for a single random variable
However, many experiments are
conducted where two random variables
are observed simultaneously to
determine not only their ind
Two Continuous Random Variables
The conditional distribution of X given a
fixed value of Y is given by:
f ( x, y )
fX|Y = y (x) =
fY ( y)
The conditional distribution of Y given a
fixed value of X is given by:
f ( x, y )
fY|X = x (y) =
f X ( x)
Example 2.
The Multinomial Distribution (sect. 3.5)
Suppose you have a situation where:
1.
You have a fixed number of trials
trials)
(n-
2. The trial are independent
However, instead of having only two
possible outcomes on each trial (success
or failure), you can ha
Note
When do you use the binomial
distribution and when do you use the
hypergeometric distribution?
You must have a situation where there
are only two possible outcomes on each
trial: success or failure
1.
If you are sampling with replacement
OR if you ar
(II) Negative Binomial Distribution
(section 3.2)
This distribution is used when trying to
determine how many trials are needed in
order to obtain r-successes
Let
X = number of trials needed to obtain the
rth success
where
1. The trials are independent
2.
Chapter 3 Special Types of Discrete
Distributions
The Binomial Distribution (section 3.1)
Bernoulli Random Variables
Let X be a random variable that can take
on only two possible values:
X = 1 if a success
0 if a failure
Let
p = P(success) = P(X = 1)
1 p
Combinations and Functions of Random
Variables (section 2.6)
1. Linear Functions
If X is a random variable and a, b are
constants, then
Y=aX+b
is also a random variable, and
(i) E[Y] = a E[X] + b
(ii) Var(Y) = a2 Var(X)
Note
An important application of th
The Gamma Distribution (section 4.3)
As we saw in the last section, an
exponential random variable describes
the length of time or distance until the
first count is obtained in a Poisson
process
A generalization of the exponential
distribution is the leng
The Exponential Distribution (section 4.2)
The exponential distribution is often used to
model failure times OR waiting times OR
interarrival times
Probability Density Function
f(x) = ex , x 0, > 0
Cumulative Distribution Function
F(x) = 1 e x
Mean and Va
The Weibull Distribution (section 4.4)
The Weibull distribution is often used to
model the time until failure of many
different physical systems.
It has the following probability density
function:
f(x) = a x
a
a 1
e
( x ) a
for x 0, a > 0, > 0
The cumulat
Chapter 5 The Normal Distribution
In this chapter we discuss the most
important special type of continuous
distribution:
The Gaussian distribution
or, as it is more commonly referred to,
The normal distribution
What Does a Normal Distribution Look
Like?
T
Normal Random Variables
1.
2.
3.
There is a probability of about 68%
that a normal random variable takes
on values within 1 standard deviation
of
There is a probability of about 95%
that a normal random variable takes
on values within 2 standard
deviatio
Case (II): Non-pooled Two Sample t-test
The underlying population standard
deviations are unknown, but are NOT
assumed to be equal
That is, x y
Substitute s2x for 2x and s2y for 2y
This leads to the following statistic:
( x y ) ( x y )
(
s
2
x
n
) +(
2
sy
(III) Dependent Samples: Inference
About the Difference Between Two
Population Means Paired t-Test
(section 9.2)
We start off with a 2-sample problem:
Xi = sample data from population1
Yj = sample data from population 2
However, subjects in our experiment
Type I and II Errors
Suppose we are performing a one-sample
t-test of
H0: = 0
HA: 0
There are 2 possible errors that can be
made:
Our
Decision
Do not
reject H0
Reject H0
True Situation
H0 is true
H0 is false
= P(Type I error)
= Prob. of rejecting H0, whe
SECTION 3.1 (PAGE 167)
R. A. ADAMS: CALCULUS
CHAPTER 3. TIONS
Section 3.1 1.
TRANSCENDENTAL FUNC-
7.
f (x) = x 2 , (x 0)
2 2 f (x1 ) = f (x2 ) x1 = x2 , (x1 0, x2 0) x1 = x2 Thus f is one-to-one. Let y = f -1 (x). Then x = f (y) = y 2 (y 0). therefore y =
SECTION 2.1 (PAGE 98)
R. A. ADAMS: CALCULUS
CHAPTER 2.
Section 2.1 (page 98)
DIFFERENTIATION
7. Slope of y =
x + 1 at x = 3 is
Tangent Lines and Their Slopes
1. Slope of y = 3x - 1 at (1, 2) is
m = lim 3(1 + h) - 1 - (3 1 - 1) 3h = lim = 3. h0 h h
4+h -
INSTRUCTOR'S SOLUTIONS MANUAL
SECTION 1.1 (PAGE 61)
CHAPTER 1.
LIMITS AND CONTINUITY
7.
Section 1.1 Examples of Velocity, Growth Rate, and Area (page 61) 1. 2.
(t + h)2 - t 2 x = m/s. t h
At t = 1 the velocity is v = -6 < 0 so the particle is moving to th
INSTRUCTOR'S SOLUTIONS MANUAL
SECTION P.1 (PAGE 10)
CHAPTER P.
Section P.1 (page 10) 1. 2.
PRELIMINARIES
19. Given: 1/(2 - x) < 3.
Real Numbers and the Real Line
2 = 0.22222222 = 0.2 9 1 = 0.09090909 = 0.09 11 Thus 99x = 12 and x = 12/99 = 4/33.
CASE I. I
INSTRUCTORS SOLUTIONS MANUAL
APPENDIX I. (PAGE A-10)
APPENDICES
15.
Appendix I. Complex Numbers (page A-10) 1. z = 5 + 2i ,
Re(z ) = 5, z = 5 + 2i z = 6 z = 4i z = i Fig. .1 x Im(z ) = 2 y z -plane
16.
17.
4 4 + 3i sin 5 5 4 |z | = 3, Arg (z ) = 5 3 and A
INSTRUCTORS SOLUTIONS MANUAL
SECTION 17.2 (PAGE 913)
CHAPTER 17. ORDINARY DIFFERENTIAL EQUATIONS
NOTE: SECTIONS 17.2 AND 17.5 AND THE REVIEW EXERCISES FOR CHAPTER 17 IN CALCULUS OF SEVERAL VARIABLES HAVE MORE EXERCISES THAN THE CORRESPONDING VERSIONS IN C
SECTION 16.1 (PAGE 858)
R. A. ADAMS: CALCULUS
CHAPTER 16. VECTOR CALCULUS
Section 16.1 (page 858) 1. Gradient, Divergence, and Curl
7.
F = x i + yj (x ) + (y) + (0) = 1 + 1 = 2 div F = x y z i j k =0 curl F = x y z x y 0 F = yi + x j (y) + (x ) + (0) = 0
SECTION 15.1 (PAGE 811)
R. A. ADAMS: CALCULUS
CHAPTER 15. VECTOR FIELDS
Section 15.1 (page 811) 1. F = x i + x j. Vector and Scalar Fields
4. F = i + sin x j.
The eld lines satisfy d x =
dy . sin x dy Thus = sin x . The eld lines are the curves dx y = cos
SECTION 14.1 (PAGE 759)
R. A. ADAMS: CALCULUS
CHAPTER 14. MULTIPLE INTEGRATION
Section 14.1 1.
Double Integrals
(page 759)
The solid is split by the vertical plane through the z axis and the point (3, 2, 0) into two pyramids, each with a trapezoidal base;
INSTRUCTORS SOLUTIONS MANUAL
SECTION 13.1 (PAGE 714)
CHAPTER 13. APPLICATIONS OF PARTIAL DERIVATIVES
Section 13.1 1. Extreme Values (page 714) 6.
1 1 < 0, and (4, 2) is a local Thus B 2 AC = 16 4 maximum. f (x , y ) = cos(x + y ), f 1 = sin(x + y ) = f 2
INSTRUCTORS SOLUTIONS MANUAL
SECTION 12.1 (PAGE 645)
CHAPTER 12. PARTIAL DIFFERENTIATION
(2,0,2)
z
Section 12.1 (page 645) 1.
f (x , y ) =
Functions of Several Variables
(2,3,2)
z=x
x+y . xy The domain consists of all points in the x y -plane not on the l
SECTION 11.1 (PAGE 597)
R. A. ADAMS: CALCULUS
CHAPTER 11. VECTOR FUNCTIONS AND CURVES
9. Position: r = 3 cos ti + 4 cos tj + 5 sin tk
Section 11.1 Vector Functions of One Variable (page 597) 1. Position: r = i + tj
Velocity: v = -3 sin ti - 4 sin tj + 5 c
SECTION 10.1 (PAGE 542)
R. A. ADAMS: CALCULUS
CHAPTER 10. VECTORS AND COORDINATE GEOMETRY IN 3-SPACE
Section 10.1 Analytic Geometry in Three Dimensions (page 542) 1. The distance between (0, 0, 0) and (2, -1, -2) is
22 + (-1)2 + (-2)2 = 3 units.
8. If A =