Exercise 1-5
Given: Three common building insulations.
Required: Thermal resistance in terms of "R" value.
Assumptions: 1. One dimensional conduction.
(i) R value of 10 cm thick berglass? Table 1.1 gives k = 0.038 W/m K, and Eq. (1.9) gives
the thermal re
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
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
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
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
Exercise 2-3
Given: A slab made up of n pairs of square bars of conductivities RA and k3, as shown.
Required: (i) Effective conductivity across the width.
(ii) Effective conductivity through the thickness.
Assumptions: 1. One-dimensional conduction.
Exercise 2-15
Given: Milk at 5C owing inside a stainless steel tube insulated with a 5 cm thick
layer of cork.
Required: Calculate the rate of heat gain per meter length of tube.
Assumptions: 1. One-dimensional steady heat ow.
2. The thennal resistance of
Fins
Ac is the cross sectional area
P is the perimeter
L Length
nf = effiency
B = (hcP/KAc)^1/2
X = BL
nf= TANHX/X
Q = h(PL)(Tb-Te)
Q = (hcP/B)(Tb-Te)TANH(BL)
Q = KaB(Tb-Te)TANH(BL)
Q = INT(0,L) (T-Te)dx
Temperature Distribution
pheta = C1e^Bx + C2e^-Bx
Chapter 1
Q = qa
Q = Delta T / Sum R
R = La / Ka
If area can be found
R = La / KaA
q = (Ti-T1)/(La/Ka) = (T2-To)/(Lb/Kb)
Q = (To - Ti) / (L/K Area)
Since its a cube not a wall
Small Grary body in nearly black surrounding
Q =(E)(sigma)A(T^4-Te^4)
When goi
(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.
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
Chapter 1 Stress
Review of Statics (1.1-1.2)
External loads
Surface force (N/m2)
concentrated force (N)
linear distributed load (N/m) (can be replaced by a resultant force)
Body force (can be represented by a single concentrated force)
Distributed Loads
MME 2202a Mechanics of Materials
Prof. L.Y. Jiang
Office:
Rm 3076 SEB
E-mail:
[email protected]
Prerequisites:
ES1022a/b/y, AM 1413 or
written permission from your Dean
Teaching Assistants:
Assignment marking:
Yang Cheng, Email: [email protected] (assign.1
Chapter 2 STRAIN
(2.1, 2.2)
Two types of strain:
Normal strain, =
Change in length resulting from applying a normal stress
unstressed length
Shear strain, = Angle of twist resulting from an applied shear stress
Normal Strain,
y =
x =
l
lo
Y
w
wo
X
l
lo
w