Calculating probabilities using the cdf
F (x) = P(X x)
Find: P(X = x)
P
(X
=
x
)
=P
=
F (
(
x
)
[X

ex
F (x
]
\

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)
[
X
<
x
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)
=
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Connection between pdf and cdf
Recall that for a continuous random variable X,
Z x
F (x) = P(X x) =
f (y)dy
1
In
9128/16
Mixed bivariate distributions
If X, Y are such that X is discrete and Y is continuous, then we say that (X, Y )
has a mixed bivariate distribution. We characterize it through the joint pmf/pdf
f (x, y)
1. f (x, y)
2.
Z
1
1
X
0 for any x, y
f (x, y
8/31/16
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Exercise 3
Consider two events A and B such that P(A) = 1/3 and P(B) = 1/2. Determ
Conditional probability
Suppose that you hold a ticket in a lottery in which six numbers are drawn without
replacement from a bin containing numbers 1 30. Your ticket shows the numbers
1, 14, 15, 20, 23, and 27. You turn on your television to watch the dr
.
1
.
STAT 6201  MATHEMATICAL STATISTICS
AUTUMN 2016
RADU HERBEI
2
.
Think probabilistically !
3
.
What is probability ?
An experiment is any process, real or hypothetical, in which the possible
outcomes can be identified ahead of time.
An event is a w
9/30/16
.
f (
x
f (a)
,
y
)

,
fdy )=

joint
marginal
f[akiy
pdf ( pmf )
pdf
)dx
of
X
f ,lx)=
f[olxiy )dy
M
Independent random variables
,
Pf
N

independent
MAN
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events
.
PIN )
Two random variables X, Y are independent if their joint cdf F (x
8/29/16
.
experiments
S
=
outcomes
sample
@ specify
space
events
0
as
cfw_
=
all
S
if
if
outcomes
possible
finite
is
5
is

S
ofonly
subset
is
event
an
infinite
countable
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if
then
subset
any
of
.
an

if
S
then
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assign
probabilities
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ALL
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STAT 6201
Homework 04
September 17, 2016
Due: in class, on Friday 09/23/2016.
Section 3.1: problem 10
Section 3.2: problem 2
Section 3.3: problems 3,5
1. A civil engineer is studying a leftturn lane that is long enough to hold seven cars. Let X
be the nu
SOLUTIONS
STAT 6201 Midterm Exam I
Ocotber 7, 2016
Name .
1. Suppose that 30 percent of the bottles produced in a certain plant are defective. If a
bottle is defective, the probability is 0.9 that an inspector will notice it and remove it
from the filling
10/3/16
Example
and
.
pdflpmf
=
jointpdflpmf
marginal
A manufacturing process consists of two stages. The first stage takes Y minutes, and
the whole process takes X minutes (which includes the first Y minutes. Suppose
that X and Y have a joint continuous
STAT 6201 Midterm Exam I
October 1, 2015
SOLUTIONS
Name .
1. (7 points) Assume that a random variable X has the following cdf
Find:
(a) Pr(0 < X 3)
(b) Pr(0 X 3)
(c) Pr(0 < X < 3)
(d) Pr(1 X 2)
(d) Pr(1 < X 2)
(e) Pr(X
5)
(f) Pr(X > 5)
=
=
Fl 3)
=
F
=
(3)
Random Variables and Distributions
Let S be the sample space associated with a probability experiment. Assume that
we have defined the events of interest and a probability assignment for each event.
A random variable is a realvalued function that is defi
Solutions
Consider
1
having
iid
n
after
after


after
1
2
n
expected
=

,
steps
,
value
distribution
the
=
n
ECX
,

X
is
position
.
1)

n
,
Xz
,
,
.
Xn
(
(
tp
particle
X.
H
1
Xi
is
Xz
+
,
is
)=n
=
,
=
of the
position
the
X
:
PCX ,
position
the
,
06
STAT 6201
Homework 07
October 23, 2016
Due: in class, on Friday 10/28/2016.
Section 4.4, problems 6,7,10,12.
Section 4.6, problem 1.
Section 4.7, problems 6.
1. Suppose that X has the uniform distribution on the interval [a, b]. Determine the mgf of X.
2.
10/10/16
Expectation
The distribution of a random variable X contains all of the probabilistic
information about X.
The entire distribution of X, however, is usually too cumbersome for presenting
this information.
Summaries of the distribution, such as
.9/19/16
discrete
,
As
rv
XE
.
distributions
&
variables
random
XE
.
[
aib
cfw_
a
X
is
discrete
we
X
is
As
.
we
,
.
XE
Xeltf
,
characterise
(
a
,a
)
)
a
X
fK)=P(X=x
Jf
.
]
to
Jf
,n
,
through
its
pmf
.
)
pdf
ft
Make
specify
its
p(a< b)
=
tattoo
Uniform di
General differentiation rules
(af + bg)0 = af 0 + bg 0 (linearity rule)
(f g)0 = f 0 g + f g 0 (product rule)
0
f
f 0g f g0
(quotient rule)
=
g
g2
(f n )0 = nf 0 f n1 (power rule)
(f (g(u)0 = f 0 (g(u)g 0 (u)u0 (chain rule)
Functionspecific differen
Problem
Az
"d
2
=
purchase
4k purchase
=
You
) RCA
Note
Pr ( A
,
:
Prl A
Pr 1 As
,
)
1
A
brand
B
is
brand
B
,
,
=PrlA
)
.
,
)
n
=
A
I
Au
/
A
,
Pr
( As
(
same
1
A Az
,
)
A Asnaa
ADR
Ain
brand
(
au
1
A
=
,
nanas
)
)
( only
D= Prl As AD
1
=
RC
Pr( A
find
I
=
Problem
Section
,
1.4
10
the
sample
5=
A=l
(
2
(
,
a
,
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B
,=lC*
C,
=L (
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't
1
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,
*
,*
,
l
three
b) cfw_
no
die
c) cfw_
at
least
d) cfw_
at
cfw_
all
most
three
,
(4
,
.
4)
,
6
*
,
(
,
*
*
,
,
't
*
,
,
*
,
,
*
*
,
)
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,
6)
outcomes
666
)
6) )
Example
Suppose that X and Y have a continuous joint distribution for which the joint pdf
is as follows:
(
x + y if 0 x 1, 0 y 1,
f (x, y) =
0
otherwise
Find E(Y
EH
X = x) and V ar(Y
)= fyfdg )dy
vary
)=E( y
'
X = x).
E(YX=x)= ygdy1x)dy
.
) ECY )2
Var(Y
.
9/12/16
probability
Conditional
A
,
B
PIAI

events
B)
p( B)
"

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>o
K
hf@f
sample
space
.
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86
PIA )

toot
PCAIB )=PfYBI
=
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100
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9.0
.
tat
)=
PCBIA
PC

BAA
)
.
It
B=
[ coating weight
is
[ surface roughness
high ]
is
high ]
a) P (
10/12/16
Expectation of functions of random variables
ynp
:
Let X be a random variable and let Y = r(X) be another random variable.
If X is a discrete r.v. with pmf f (x) then
E(Y ) = E(r(X) =
X
r(x)f (x)
all x
If X is a continuous r.v. with pdf f (x) t
Prediction
#
SKIP
Suppose X, Y are random variables and the goal is to predict one of them, say Y .
Let d be the predicted value.
If no information is available, the prediction that minimizes the mean squared
error (MSE)
MSE = E (Y d)2
among all possible
Random Variables and Distributions
( 5
,
events
PC )
.
,
]
Let S be the sample space associated with a probability experiment. Assume that
we have defined the events of interest and a probability assignment for each event.
A random variable is a realvalu
STAT 6201
Homework 01
August 29, 2016
Due: in class, on Friday 09/02/2016.
Section 1.4: problems 10, 11.
Section 1.5: problem 11
1. Three sixsided dice are rolled. The six sides of each die are numbered 1 6. Let A be the
event that the first die shows an
A box contains three coins with a head on each side, four coins with
a tail on each side, and two fair coins. If one of these nine coins is
selected at random and tossed once, what is the probability that a
head will be obtained?
the
Define
T
,
selected
=
09/21/16
Example
Suppose that a random variable X can take only the values 2, 0, 1, and 4, and that
the probabilities of these values are as follows:

P(X =
2) = 0.4 P(X = 0) = 0.1 P(X = 1) = 0.3 P(X = 4) = 0.2
Find and sketch the cdf of X.

FCX
)=P(
Xs
Covariance and correlation
Consider two random variables X, Y , and assume that they are dependent.
A quantitative measure of association between X and Y is the covariance or
the correlation. Let E(X) = X and E(Y ) = Y . Then,
Cov(X, Y ) = E (X X )(Y Y
9/26/16
.
P(
P
(
X<
2)
11/11/1111
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region
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tools
region
region
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interest
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)
region
R
~
11/11/11
P(
region
)=
)fK)dx
region
P(
1<
XE
5)
=
Jfk
( 1,5 ]
)dx
=) ,5fk)dx
2)
.
cdf
F(x)=
f(x)=F'
Hotly )ay
( )
x
a

)=P(Xex)
F
STAT 6201
Homework 03
September 11, 2016
Due: in class, on Friday 09/16/2016.
Section 2.1: problems 4, 14
Section 2.2: problems 12, 16
1. Each time a shopper purchases a tube of toothpaste, he chooses either brand A or brand
B. Suppose that for each purch
10/19/16
Example (important)
"%
My
Suppose that X has a continuous distribution with pdf given by
f (x) = ce
(x 3)2 /2
1<x<1
Determine the mean of X and all the central moments.
fire
3
30
El. X )
E ( )
fjxfk)dx
=
.
=
'
Fox .ci
=c[yx
y=x
[g.
=
"
c.
e
"0x