STAT 516Spring 2013
Practice Midterm: Probability and Distributions
Name:
Please return this page with your solution after exam.
1. Five cards are drawn at random from a 52-card deck.
(a). Compute the probability that at least two of them are spades.
(b).
STAT 516Spring 2013
Practice Final Exam
Name:
1.
Let X1 and X2 have the joint probability mass function described by the following
table:
(x1 , x2 ) (0, 1) (0, 2)
p(x1 , x2 )
0.1
0.2
(1, 1)
0.3
(1, 2)
0.2
(2, 1)
0.1
(2, 2)
0.1
(a). Find P (X1 + X2 3|X1 1)
I NTRODUCTION TO P ROBABILITY T HEORY
Conditional Probability
Conditional Probability: of event C2 given the event C1 , provided that
P (C1 ) > 0,
P (C1 C2 )
P (C2 |C1 ) =
P (C1 )
Properties:
P (C2 |C1 ) 0
P (C2 C3 |C1 ) = P (C2 |C1 ) + P (C3 |C1 ) + ,
Chapter 2: Multivariate Distributions
Distribution of Two Random Variables
Transformations: Bivariate Random Variables
Conditional Distributions and Expectations
The Correlation Coefcient
Independent Random Variables
Extention to Several Random Variables
M ULTIVARIATE D ISTRIBUTIONS
Notation
Let X and Y have joint pdf f (x, y ).
2
2
1 and 2 are the means of X and Y , 1 and 2 are the variances of
X and Y .
u(x, y ) is a function of x and y .
M ULTIVARIATE D ISTRIBUTIONS
Covariance and Correlation Coefci
I NTRODUCTION TO P ROBABILITY T HEORY
Expectation
Expectation: Let X be a random variable. If X is a continuous
random variable with pdf f (x) and |x|f (x)dx < , then the
expectation of X is
E (X ) =
xf (x)dx.
If X is a discrete random variable with pmf
I NTRODUCTION TO P ROBABILITY T HEORY
Important Inequalities
Theorem 1.10.1: Let X be a random variable and let m be a positive
integer. Suppose E [X m ] exists. If k is an integer and k m, then
E [X k ] exists.
Theorem 1.10.2: (Markovs Inequality). Let
I NTRODUCTION TO P ROBABILITY T HEORY
-eld
Interested in assigning probabilities to events, complements of events,
and union and intersections of events.
We want our collection of events to include these combinations of
events.
-eld: Let B be a collec
STAT 516
Practice Final Exam
Name:
1. Suppose that X and Y are two bernoulli random variables. Prove that X and Y are
independent if they are uncorrelated: Cov(X, Y ) = 0.
2. Let X1 and X2 be independent normal random variables with distribution N (1, 1)
STAT 516
Practice Midterm: Probability and Distributions
Name:
Please return this page with your solution after exam.
1. Five cards are drawn at random from a 52-card deck.
(a). Compute the probability that at least two of them are spades.
(b). What is th
M ULTIVARIATE D ISTRIBUTIONS
Independent Random Variables
Let X1 and X2 denote the random variables of continuous type which
has joint pdf f (x1 , x2 ) and marginal pdf f1 (x1 ) and f2 (x2 ),
respectively.
Conditional pdf f2|1 (x2 |x1 ): f (x1 , x2 ) =
M ULTIVARIATE D ISTRIBUTIONS
Review
We introduced the joint probability distribution of a pair of random
variables.
Recover the marginal distributions from the joint distribution.
In this section, we discuss the conditional distributions, i.e., the
dis
M ULTIVARIATE D ISTRIBUTIONS
Motivation
Let (X1 , X2 ) be a random vector. Suppose we know the joint
distribution of (X1 , X2 ) and we seek the distribution of a
transformation of (X1 , X2 ), say, Y1 = u1 (X1 , X2 ) and
Y2 = u2 (X1 , X2 ).
We know how t
I NTRODUCTION TO P ROBABILITY T HEORY
Mean
If the support of X is cfw_a1 , a2 , , it follows that
E (X ) = a1 p(a1 ) + a2 p(a2 ) + a3 p(a3 ) + .
This sum is seen to be a weighted average of the values of
a1 , a2 , . . . This suggests that we call E (X )
Chapter 1: Probability and Distributions
Introduction
Set Theory
The Probability Set Function
Conditional Probability and Independence
Random VariablesDiscrete and Continuous
Expection of a Random Variable
I NTRODUCTION TO P ROBABILITY T HEORY
Introductio
I NTRODUCTION TO P ROBABILITY T HEORY
Random Variable
Random Variable: Consider a random experiment with a sample
space C . A function X , which assigns to each element c C one and
only one number X (c) = x, is called a random variable. The space
or rang
I NTRODUCTION TO P ROBABILITY T HEORY
Discrete Random Variable
Discrete Random Variable: its space is either nite or countable
A set D is countable if its elements can be listed; i.e., there is a
one-to-one corresponding between D and the positive integ
I NTRODUCTION TO P ROBABILITY T HEORY
Continuous Random Variable
Continuous Random Variable: if its cumulative distribution function
FX (x) is a continuous function for all x R.
If X is continuous then P (X = x) = FX (x) FX (x) = 0.
x
FX (x) =
fX (t)dt