Chapter 4: Bivariate Distributions
STAT2001
2010 Term I
Outline
1. Two Discrete Random Variables
2. Two Continuous Random Variables
3. Independence
4. Expectation
5. Covariance and Correlation
(Textbook chapters: 4.1 - 4.3)
1
1. Two Discrete Random Variab
STA2001 Tutorial 9
Nov. 25, 2010
1. Random variables X and Y have p.d.f.
2exy ,
0,
fX,Y (x, y ) =
0<x<y<
otherwise
(a) Determine the marginal p.d.f of X and Y
(b) Calculate Cov(X,Y)
(c) Are X and Y independent?
(d) Calculate Var(X+Y)
Sln:
(a)
2exy dy = 2e
STA2001 Tutorial 10
Dec. 2, 2010
1. Suppose that X and Y are independent random variables with the same
exponential density
1
f (x) = ex/ ,
x > 0.
Show that W = X + Y and Z = X/Y are independent. (2005 Final 3)
2. X1 , .Xn are independent Gamma random var
STA2001 Tutorial 8 Solution
Nov. 11, 2010
1. Let X have a uniform distribution U (0, 2), and the conditional distribution of Y given X=x, is U (0, x2 )
(a) Determine the joint p.d.f. of X and Y, fX,Y (x, y )
(b) Calculate fY (y ),the marginal p.d.f. of Y
STA2001 Tutorial 3
Oct. 7, 2010
1. Alternative ways of understanding sampling
Ordered
Unordered
w/ replacement
w/o replacement
nr
n(n 1).(n r + 1)
n Cr
(n+r1) Cr
Theorem 1 (Binomial Series Theorem)
For any real number and |x| < 1,
1
1
()( 1)x2 + ()( 1)(
STA2001 Tutorial 6
Oct. 28, 2010
Attention: Question 3 is modied to avoid ambiguity. Students who have
attended the tutorials, especially at the rst time slot, please refer to the
explanation in the solution.
1. Trac on a road is passing a point at rate o
STA2001 Tutorial 10
Dec. 2, 2010
1. Suppose that X and Y are independent random variables with the same
exponential density
f (x) =
1
exp(x/),
x > 0.
Show that W = X + Y and Z = X/Y are independent. (2005 Final 3)
ZW
Z +1
Solution: While X =
J=
X
Z
Y
Z
X
STA2001 Tutorial 9
Nov. 25, 2010
1. Random variables X and Y have p.d.f.
fX,Y (x, y ) =
2exy ,
0,
0<x<y<
otherwise
(a) Determine the marginal p.d.f of X and Y
(b) Calculate Cov(X,Y)
(c) Are X and Y independent?
(d) Calculate Var(X+Y)
2. A consulting actua
STA2001 Tutorial 8
Nov. 11, 2010
1. Let X have a uniform distribution U (0, 2), and the conditional distribution of Y given X=x, is U (0, x2 )
(a) Determine the joint p.d.f. of X and Y, fX,Y (x, y )
(b) Calculate fY (y ),the marginal p.d.f. of Y
(c) Compu
STA2001 Tutorial 6
Oct. 28, 2010
Attention: Question 3 is modied to avoid ambiguity. Students who have
attended the tutorials, especially at the rst time slot, please refer to the
explanation in the solution.
1. Trac on a road is passing a point at rate o
Chapter 2: Discrete Distributions
STAT2001
2010 Term I
Outline
1. Discrete random variables
2. Mathematical expectation
3. Binomial distribution and Hypergeometric distribution
4. Moment generating function
5. Poisson distribution
(Textbook chapters: 2.1
Chapter 3: Continuous Distributions
STAT2001
2010 Term I
Outline
1. Continuous random variables
2. The Uniform distribution and Exponential distribution
3. The Gamma distribution and Chi-square distribution
4. The Normal distribution
(Textbook chapters: 3
Chapter 1: Fundamental Concepts
STAT2001
2010 Term I
Outline
1. Random experiments and Sample Space
2. Probability
3. Methods of enumeration (Counting)
4. Conditional probability
5. Independent events
6. Bayess theorem
(Textbook chapters 1.1 - 1.6)
1
1. R
Chapter 5: Distributions of functions of random variables
STAT2001
2010 Term I
Outline
1. Functions of one random variable
2. Transformations of two random variables
3. Several independent random variables
4. Random functions associated with Normal distri
STA2001 Tutorial 4
Oct. 14, 2010
1. Revision
Discrete Random Variable
Discrete random variable takes on nite or a countable innite
number of values.
Probability Mass Function (p.m.f.)
f (x) = P (X = x), P (X A) =
f (x) = 1
f (x),
x S
xA
Cumulative Dist
STA2001 Tutorial 2
Sept. 30, 2010
1. Counting
Multiplication Rule
E1 has n1 outcomes,E2 has n2 outcomes E1 E2 has n1 n2 outcomes
Sampling
w/ replacement
w/o replacement
r
Ordered
n
n(n 1).(n r + 1)
Unordered
Cr
n Cr
(n+r1)
Examples
(a) CUHK will hold a
STA2001 Tutorial 4 Solution
Oct. 14, 2010
(1) In NBA, a shooting guard has 40% eld goal percentage. In a specic
game,
(a) What is the probability he has the rst eld goal in the 5th shot?
Sln:
P (A) = 0.64 0.4 = 0.05184
(b) What is the probability he has t
STA2001 Basic Concepts in Statistics and Probability I
Tutorial 3
Oct 20
. Bag
B1 contains two red and four white balls; bag Bontains one red and
c2
two white balls; and bag B3 contains five red and four white balls.
The
probabilities for selecting the ba
STA2001 Tutorial 1
Sept. 16, 2010
1. Sample space
(a) The sample space is a collection of possible outcomes of an experiment. For example, we have S1 = cfw_H, T for tossing a coin,
S2 = cfw_1, 2, 3, 4, 5, 6 for rolling a dice.
(b) Question 1: How about t