MA2216/ST2131 Probability
Notes 6
Joint Probability Distributions
So far, we have only considered probability distributions for single random variables. There will be situations, however, where we may nd it
desirable to record the simultaneous outcomes of
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 1
1. A die is rolled continually until a 6 appears, at which point the experiment stops.
(i) What is the sample space of this experiment?
(ii) Let En de
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 2
1
1. Let = cfw_1, 2, 3, 4 and to each point be assigned probability 4 . Let
A = cfw_1, 2,
B = cfw_1, 3,
C = cfw_1, 4.
(i) Prove that A, B , and C are
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 4
1. An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of
them are white and 2 are black, we stop. If not, we replace the ball
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 5
1. Find the number of distinct positive integer-valued vectors (x1 , x2 , . . . , xr ) satisfying
x1 + x2 + + xr = n,
xi > 0, i = 1, . . . , r.
HINT:
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 6
1. The joint probability density function of X and Y is given by
f (x, y ) = e(x+y) , 0 x, y < .
(a) Find the marginals fX and fY , respectively.
(b)
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 7
1. Let X1 , X2 , . . . , Xn be i.i.d. normal random variables. Say, the common distribution
is N (, 2 ). Show that there are constants An and Bn such
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 8
1. Let X and Y be independent uniformly distributed random variables over (0, 1].
Dene
U = 2 log X cos(2Y )
V = 2 log X sin(2Y ).
Find the joint p.d.f
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 9
1. Let X be uniformly distributed over (0, 1). Let Y = 2 ln(X ).
(i) Find the probability density function of Y .
(ii) Do you recognize the distributi
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 10
1. Let X be the number of 1s and Y the number of 2s that occur in n rolls of a fair
die. Let
cfw_
cfw_
1, if the ith roll lands on 1,
1, if the i rol
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 11
1. Suppose that the expected number of accidents per week at an industrial plant is 5.
Suppose also that the number of workers injured in each accide
MA2216/ST2131 Probability
Notes 1
A Brief Historical Account.
About three hundred and sixty years ago, it is said, correspondences
(in the 1650s) between two famous French mathematicians, Blaise Pascal
(1623-62) and Pierre de Fermat ( 1601-65), gave birth
MA2216/ST2131 Probability
Notes 2
1. Axioms of Probability
Consider an experiment whose sample space is S .
The objective of probability is to assign to each event A a number
IP(A), in [0, 1], called the probability of the event A, which will give a
preci
MA2216/ST2131 Probability
Notes 3
1. Random Variables.
It is frequently the case that, when an experiment is performed, we are
mainly interested in some function of the outcome as opposed to the
actual outcome itself.
For instance, in testing 100 electron
MA2216/ST2131 Probability
Notes 4
1. Discrete Uniform Distribution.
1. If X assumes the values x1 < x2 < . < xn , with equal probabilities,
then X is called a discrete uniform r.v. The probability distribution
of X is called the discrete uniform distribut
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 11
Solution Set
1. Sol. Let N denote the (random) number of accidents per week at that industrial
plant. By assumption, E [N ] = 5. Denote by Xn the num
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 10
Solution Set
1. Sol. Observe rst that for part (i),
IP cfw_Xi = 1 = IP cfw_Yi = 1 =
1
.
6
Note also that Xi Yi = 0 for all i.
(ii) For i = j ,
IP cfw
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 9
Solution Set
1. Sol. First, observe that Y is a positive real-valued r.v. To determine the distribution of Y , we begin with nding its c.d.f. For y >
MA2216/ST2131 Probability
Notes 7
Sums of Independent Random Variables
Very often we are interested in the sum of independent random
variables. When X and Y are independent, we would like to know the
distribution of X + Y . In the following, we will deal
MA2216/ST2131 Probability
Notes 8
Multidimensional Changes of Variables
and
Bivariate Normal Distribution
1. Changes of Variables
Let X1 and X2 be jointly distributed random variables with joint probability density function fX1 ,X2 . It is sometimes neces
MA2216/ST2131 Probability
Notes 9
Review, Examples
and
Properties of Expectation
1. Linear Transformation.
Let X1 , X2 , . . . , Xn be continuous random variables having joint density
f and let random variables Y1 , Y2 , . . . , Yn be dened by the followi
MA2216/ST2131 Probability
Notes 10
Properties of Expectation
and
Conditional Expectation
1. Summary of Basic Properties.
Let us rst review some elementary properties of mathematical expectation, and then develop and exploit additional properties and usefu
MA2216/ST2131 Probability
Notes 11
Moment Generating Functions
and
Inequalities
1. Moment Generating Function.
1. Recall the denition of MGF (moment generating function) of X (which
had been mentioned in 4.7, Notes 3):
The moment generating function of X
MA2216/ST2131 Probability
Notes 12
Central Limit Theorem
1. Convergence in Distribution.
1. Denition. Let W1 , W2 , . . . be a sequence of random variables having
distribution functions FWn , respectively. W1 , W2 , . . . are said to converge in distribut
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 1
Solution Set
1. Sol. The sample space S is the union of the following two sets:
cfw_(x1 , x2 , . . . , xk ) : xi = 6 for i = 1, 2, . . . , k 1; xk = 6
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 2
Solution Set
1
. Also,
2
AB = AC = BC = ABC = cfw_1.
11
1
Thus, = IP(AB ) = IP(AC ) = IP(BC ) = , which establishes (i).
4
22
Note. That A and B are m
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 3
Solution Set
1. Sol. First of all, observe that one should make use of Bayes rule. Fix i. Let Bk be
the event that the ball is in the k th box, where
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 4
Solution Set
1. Sol. Note that the sampling is with replacement. The underlying model becomes
a geometric one with parameter p =
to p (1 p)4 .
4
2
4
2
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DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 5
Solution Set
(
)
n1
1. Sol. Follow the hint, which gives the answer right away. That is, there are
r1
distinct positive integer-valued vectors (x1 , x
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
Year 2012/13
MA2216/ST2131 Probability
Tut. 6
Solution Set
1. Sol. First of all, observe that f (x, y ) = ex ey for all 0 < x, y < , we conclude
that X and Y are independent and both have an expon