xt
1 e e
0
x
dx
= ? i.e.,
1
x(t )
1 e dx
0
=?
(Scientific American) A man is mugged and claims his mugger was black. However, when the
scene was reenacted many times, the man correctly identified the race of his attacker only 75% of
the time. What is t

Section 5.6 (Normal Distribution)
Theorem 1 (Recall): Write it down. You will need it.
2
1 x
(
)
1
e 2 dx
2
= 1 = 100%
Definition 1: Do you need to copy this?
Let X be a random variable. If
2
f(x) =
1 x
(
)
1
e 2
,-
2
< x < , and are fixed, - < <

You need to bring the trimmed version to the next class.
This will be counted as part of your HW grade.
Section 4.5 (Geometric random variable)
Recall: If |a| < 1, then |an+1| 0 when n
Lemma 1 (recall):
If |a| <1, then
a k1
k=1
1
= 1 + a + a2 + a3 + = 1

Section 4.4 (The Binomial Distribution)
Example 4:
(a) A baking machine blends 1,000 raisins into a dough mix and from this makes 500
cookies. Randomly pick a cookie. Let Y = # of raisins in your cookie.
E(Y) = _.
SD(Y) = _.
P[Y > 1] = _.
Solution:
Y is B

Section 4.10 (The Moment-Generating Function)
Semi-Quiz: Thumb up if you are ready.
X =p
1) If X is Bernoulli, then
X = pq
X =np
2) If X ~ B(n, p), then
X = npq
how do prove? Divide and Conquer.
You will need this again in Section 4.10 and Chapters 5

Section 4.7 (Poisson Approximation for Rare Event)
Google Translate > Poisson (from French to English) fish
It is Poisson, not poison.
In this course, Poisson distribution and Poisson probability are referred to the
works and the legacy of Simon Denis Poi

HW:
#A: Test correction
#B: Review:
a) Prove that
n
n
i=1
i=1
a x i=a x i
n
hint: expand the summation
n
(xi )= x in
b) Prove that
i=1
i=1
#C: Let X = Uniform[0,1]. Find and plot the density function of Y.
(a) Y = X2
(b) Y = X3
#D: Let X1, X2, ., Xn be

Test alert:
a) We will have Exam-1 on Thursday, 9/25.
b) You are allowed to prepare formula sheets (one-sided, 2 pages limit). You
can copy formulas but you are not allowed to copy any proofs or examples.
c) There will be proofs and Calc Review problems i

Discrete version (old HW):
e.g.,
s
#B: p.224 #47, 49, 57,
Hand-in #B:
a)
x 36 ex dx
0
=?
b)
x 9 /2 ex dx
0
= ? Gamma(11/2) = (9/2)*(7/2)*(5/2)*(3/2)*(1/2)*Gama(1/2)
= 52.34
c)
x 17 ex dx
0
=?
Old HW:
2
1 x
(
)
1
e 2 dx
2
= 1 = 100%
Hint (first step)

Solution 2:
(n2)
=
n!
2 ! ( n2 ) !
=
n(n1)
2
= Solution-1.
Guidelines on HW and Group Meeting (sent to you yesterday):
a) In certain groups, some students got the correct answer, while others did not.
Apparently their group meeting did not work the way it

Group names: _, _, _, _, _.
Put your name/group names on the top of Page 1 of your hand-in HW.
You will get bonus if the book answer if wrong.
If you think the book answers are wrong on certain problems, you need to spell
them out on the top of p.1 of yo

Section 6.5
(Conditional Expectation)
Goal: E[E(Y|X)], double expectations and applications
Example-1 (Recall-1, #24, p. 312)
cfw_
f ( x , y )= x + y , 0 x , y 1
0, elsewhere
Mathematica [Section 6.5 (Conditional Expectation).nb]
1
f X (x)
(a)
1
f (x , y

Section 5.4 (Exponential Variable)
Semi-quiz:
1)
d ax
e =a e ax
dx
2)
e ax dx
3)
e x/ dx
4)
x
5)
x(t )
1
e dx
0
1
= ae
1
lim e x
ax
+c
x /
=- e
+c
e =
= 0 [Think:
1
1
=
1
=
x(t )
1
e
t 1
=
1
1t , if
1
1
t
0 =
t<
e
1
e
= 0]
1
x(t )
0
1
e - e 0 ] if

Section 5.3 (Uniform Random Variable)
Example 1:
Randomly pick a number from [0, 1] = cfw_x: 0 x 1 .
Let X = the outcome of your pick.
Mathematica:
RandomReal []
RandomReal [1,5]
P[0.0 X 0.1 ] = 10%
P[0.1 X 0.2 ] = 10%
P[0.2 X 0.3 ] = 10%
P[0.7 X 0.8 ] =

Mat-316 (Lecture-2)
a) Name card
b) Benefit of Cooperative Learning
Benefits of Cooperative Learning
(A rewriting largely based on Reynolds et al., 1995)
On the left margin, mark your evaluation of each item:
-10 (strongly disagree), -5 (disagree), etc.,

Semi-quiz:
Theorem 2: (the no-memory property)
If X is Exp( ), then P[X > a+b|X > a] = P[X > b]
Theorem 3:
Given t, let
Xt
= # of occurrences of certain event in the time interval [0, t] Student-1
W = waiting time until this event happens (e.g., a traffic

Previous HW:
Do you prefer your instructor do the following on the board?
Negative Binomial:
Do you prefer your instructor do the following on the board?
How many of you want me to send the HW keys to you?
I will not teach M-316 next semester but would co

Section 8.2 (Convergence in probability)
Heads and Tails problem:
Probability of heads: p, with 0 p 1
Question: Would we expect the estimate to fall closer to p for larger
sample sizes? Intuiton tells so.
Question: Is it always true that larger sample siz

a
*
Letter-envelop problem: P[X2 = 1] = 1/n
Example 3:
X and Y are independent.
X
= 5, X = 6
Y
a)
= 7, Y = 4
2 X +3 Y
= 10+ 21 = 31
2 X +3 Y
b)
2 X6
2 X 6
c)
= 10 6 = 4
=
2 X3 Y +9
2 X 3Y + 9
d)
1
2
1
2
122+122 (Use Geometry) = 16.97
=
( X +Y 3 )
(X +

Lecture-1
Windows:
1) PAWS
2) Math-316 folder:
Syllabus: trimed and long versions
Lecture-1
3) Google
1. Name cards (Free bookmark. Dont lose it.)
2. Syllabus
3. I will send the entire lecture notes to you later today. i.e., you dont have to copy anythi

Old HW:
1. (Fish in the Lake): How would you estimate the number of fish in a lake? Statisticians have devised
the so-called capture-recapture method to solve this and similar problems. A first group of fish is
caught, tagged, and released. A few days lat

Previous HW:
Book errors:
#4.69(a): 8.85%
#4.75(a): 10.5%
#4.89(d): 40.86%
#4.93(a, b): 14.9%, 2.96%
Bonus: 5 points (but you need to spell them out on p.1 of your HW as said in an
earlier class meeting).
Example 2:
On I-95, New Jersey Turnpike, Johnny dr

Recall:
Let X be a random variable. If
2
f(x) =
1 x
(
)
1
e 2
,-
2
< x < , and are fixed, - < <
, > 0,
then X is called a Normal( , ) random variable, or Gaussian in high level
scientific applications.
Section 7 .3 (Jacobian and Monte Carlo)
Google >

Section 6.1 (Bivariate Random Variables)
Semi-quiz:
lim x
x/2
lim x e
=
x
x
x/ 2
e
lim 1
x
= ( 1 )e x/ 2 = 0.
2
Hand-in #A:
.
Hint: .
Example 1:
(a) From Stat-215:
(b) Re-format the table: Write it down.
Student(1)
y
Staff(2)
f X (x)
American(1)
x
Europe

One student sent me the following:
Is the formula correct?
Yes, and I like it. Yummy, yummy.
If you have something similar, send to me for bonus credits.
*
Old HW:
Cov(X, -3X+4) = E[(X- X (3 X + 4cfw_3 X + 4 ) = (-3)E[(X- X )(X- X
=
-3Var(X).
*
Section