chord with C as midpoint, 3
1
l
if and only if C lines inside the circle with center O and radius
2 1
. Thus 4
1 2
1
2
A P
. Solution 2. Because of symmetry, we may fix one endpoint of the chord at some point
P
and then choose the other endpoint
1
P at random. Let the probability that
1
P lies on an arbitrary arc of the circle be proportional to the length of this arc. Now the inscribed equilateral
triangle having
P
as one of its vertices divides the circumference into three equal parts. A chord drawn through
P
will be longer than the side of the triangle if and only if the other endpoint
1
P of the chord lies on that one-third of the circumference that is opposite
P
. It follows that the required probability is
3 1 .
Solution 3. Note that the length of a chord is determined uniquely by the distance of its midpoint from the center of the circle. Due to the symmetry of the circle, we assume that the
midpoint of the chord lies on a fixed radius, OM , of the circle. The probability that the midpoint
M
lies in a given segment of the radius through
M
is then proportional to the length of this segment. Clearly, the length of the chord will be longer than the side of the
inscribed equilateral triangle if the length of OM is less than 2
radius . It follows that the
required probability is 2
1 .
Question: What ’s happen? Which answers is are right?
Example: Consider sampling
2
r
items from 3
n
items, with replacement. The outcomes in the ordered and unordered sample spaces are these.
Unordered {1,1}
{2,2} {3,3}
{1,2} {1,3}
{2,3} Probability
16 16
16 16
16 16
Ordered 1,1
2,2 3,3
1,2, 2,1
1,3, 3,1
2,3, 3,2
Probability 19
19 19
29 29
29 Which one is correct?
Hint: The confusion arises because the phrase “with replacement” will typically be
interpreted with the sequential kind of sampling, leading to assigning a probability 29 to the event {1, 3}.
D. Conditional Probability and Independence
Definition
Conditional probability of
A
given
B
is
| B
P B
A P
B A
P
,
provided
B
P
.
Remark: a In the above definition,
B
becomes the sample space and
1 |
B
B P
. All events are calibrated with respect to
B
. b If
B A
then
B
A P
and
| |
A B
P B
A P
. Disjoint is not the same as independent.
Definition
A
and
B
are independent if
| A
P B
A P
. or
A P
B P
B A
P
Example: Three prisoners,
A
,
B
, and C , are on death row. The governor decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his
choice but requests that the name be kept secret for a few days. The next day,
A
tries to get the warden to tell him who had been pardoned. The warden refuses.
A
then asks which of
B
or C will be executed. The warden thinks for a while, then tells
A
that
B
is to be executed.
Warden ’s reasoning: Each prisoner has a 13 chance of being pardoned. Clearly, either
B
or C must be executed, so I have given
A
no information about whether
A
will be pardoned.
A
’s reasoning: Given that
B
will be executed, then either
A
or C will be pardoned. My chance of being pardoned has risen to 12.
Which one is correct?
Bayes ’ Rule
,
,
2 1
A A
: partition of sample space,
B
: any set,
|
| |
j j
i i
i i
A P
A B
P A
B P
A P
B P
B A
P B
A P
. Example: When coded messages are sent, there are sometimes errors in transmission. In
particular, Morse code uses “dots” and “dashes”, which are known to occur in the proportion
of 3:4. This means that for any given symbol,
7 3
sent
dot P
and
7 4
sent
dash P
. Suppose there is interference on the transmission line, and with probability 18 a dot is
mistakenly received as a dash, and vice versa. If we receive a dot, can we be sure that a dot was sent?
Theorem
If
B A
then a
c
B A
, b
B A
c
, c
c c
B A
. Definition
n
A A
, ,
1
: mutually independent if any subcollection
ik i
A A
, ,
1
then
k j
ij k
j ij
A P
A P
1 1
. E.
Random Variable Definition
Define
S X
: new sample space
.
X
: random variable,
S
X :
, ,
,
X
P P
S
, where
X
P : induced probability function on
in terms of original
P
by }
: {
i j
j i
X
x s
X S
s P
x X
P
,
and
X
P satisfies the Kolmogorov Axioms.
Example: Tossing three coins,
X
: of head S =
{HHH ,
HHT, HTH,
THH, TTH,
THT, HTT,
TTT} X :
3 2
2 2
1 1
1
Therefore,
} 3
, 2
, 1
, {
, and
8 3
} ,
, {
} 1
: {
1
HTT
THT TTH
P s
X S
s P
X P
j j
X
. F.
Distribution Functions
With every random variable
X
, we associate a function called the cumulative distribution function of
X
.
Definition
The cumulative distribution function or cdf of a random variable
X
, denoted by x
F
X
, is defined by x
X P
x F
X X
, for all
x
. Example: Tossing three coins,
X
: of head
3 ,
2 ,
1 ,
X
, the corresponding c.d.f. is
x if
x if
x if
x if
x if
x F
X
3 1
3 2
8 7
2 1
2 1
1 8
1 ,
where
X
F : a is a step function
b is defined for all
x
, not just in
} 3
, 2
, 1
, {
c jumps at
i
x , size of jump
i
x X
P
d
x
F
X
for
x ;
1
x F
X
for 3
x
e is right-continuous is left-continuous if x
X P
x F
X X
Theorem
x F
is a c.d.f. a
lim
x F
x
,
1 lim
x F
x
.
b
x F
: non-decreasing c
x F
: right-continuous Example: Tossing a coin until a head appears. Define a random variable
X
: of tosses required to get a head. Then
p p
x X
P
x 1
1
,
,
2 ,
1
x
,
1
p
. The c.d.f. of the random variable
X
is
x X
p x
X P
x F
1 1
,
,
2 ,
1
x
. It is easy to check that
x F
X
satisfies the three conditions of c.d.f. Example: A continuous c.d.f. of logistic distribution is
x X
e x
F
1 1
, which satisfies the three conditions of c.d.f.
Definition
a
X
is continuous if x
F
X
is continuous. b
X
is discrete if x
F
X
is a step function.
Definition
X
and
Y
are identical distributed if
A ,
A Y
P A
X P
. Example: Tossing a fair coin three times. Let
X
: of head and
Y
: of tail. Then
k Y
P k
X P
,
3
, 2
, 1
,
k
. But for each sample point
s ,
s Y
s X
.
Theorem
X
and
Y
are identical distributed
x F
x F
Y X
,
x
.
F. Density and Mass Function Definition
