APPLIED MATHEMATICS 30063
Review exercises
2. Probability
1. The sample space of a certain experiment is
= f1;2;3g
a) Find the algebra of the power set P() and verify it has 23= 8 elements.
b) Find the smallest algebra Acontaining E=f1;2gand verify that A$P().
c) If P(E) = 0:5, …nd the other values the probability measure P:A ! Rtakes on the elements
of the algebra A.
2. Let Aand Bbe two events. If P(A)=0:6;P(B)=0:5;P(AjB) = 0:4…nd
a) P(BjA);
b) P(A\B);
c) P(A[B);
d) P(AnB)(AnB=A\Bis the set-di¤erence of Aand Band consists of the elements of Athat
do not belong to B);
e) P(BnA).
3. Let A=?;; E; E be an algebra of events where with E=fa;b; cgand E=fd; eg.
a) Find .
b) Let X: !Rbe such that X(a) = 20000 ;X(d) = 40000. Find the values Xtakes on b;c
and esuch that Xis B-measurable with respect to A.
c) Suppose Xis B-measurable and P(E)=0:75. Find the probability function pand the distrib-
ution function FXof X.
4. Let Xbe a continuous random number such that
P(0 <X1) = 0:2
P(1 <X5) = 0:5
P(5 <X10) = 0:3
Suppose Xis uniformly distributed on each of the 3 intervals above.
a) Find the probability density of X.
b) Find E (X).
c) Find (X).
5. Consider the discrete random vector X=X1
X2with joint probability mass function
X2X12 4 6
10 0:1 0:2 0:3
20 0:2 0:1p
a) Find pand the marginal probability functions PX1and PX2of X1and X2.
b) Find P(X1= 4jX2= 10) and P(X14jX2= 20).
c) Find Cov(X1;X2). Are X1and X2stochastically dependent?
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