Exercise 11 :
A child collect soccer player photos, that he nds in cereal boxes. Each cereal
box contain one photo. Each photo is equally likely to be found in a cereal box. The
complete collection co
Fall 2015 Math 130A
Review problems
Exercise 1 : Let X be an uniform random variable over [ a, b] with 0 < a < b. Let
Y = X2.
(a) Compute E(Y ) and Var(Y ).
Answer :
b3
3( b
a3 4( b a )4
,
.
a)
45
(b)
2
n
n!
Answer : Let n = 2k.
.
k
2k
Exercise 17 :
In a calling center, a client has a probability 0.25 to wait until someone picks up.
(a) A client calls 4 times. Let X be the number of calls where he
X
The probability that the satellite system will function corresponds to the event
k. We write the law of total probability
n
n
n
P( X k) = P( X = i) =
P( X = i | R) P( R) + P( X = i | Rc ) P( Rc )
Let X be the number of Heads that we obtain when we flip a coin n times. We
have X B(n, p). The probability to get an even number of Heads is then
b n2 c
b n2 c
n 2i
P(X is even) = P( X = 2i ) =
p (
We consider the sum
n
k p k (1
k =0
n
k p k (1
k =0
p)n
p)n
k k
x . By the binomial theorem we have
k k
p)n .
x = ( xp + 1
We integrate both sides to get
n
k p k (1
k =0
p)n
k
x k +1
( xp + 1
Fall 2015 Math 130A
Homework solutions
Due Date
Sections
Problems
Thu, Nov. 12
Chapter 4 - Random Variable
71 , 72 , 78, 79
Theoretical : 25, 30 , 35
Exercise 71 :
(a) Let X be the number of bets that
We make the change of variable l = k + 1 in the last sum to obtain
k
n =
k=n
N
N +1
l = n +1
1
l
n
Therefore we get
E (Y ) =
N +1
=
l = n +1
+1
n( N
n +1 )
( Nn )
=
l 1
n+1 1
=
N+1
.
n+1
n ( N + 1)
Theoretical exercises
Exercise 25 : Let Y be the number of events that occur. Let X be the number of events
that are counted. Let n 2 N. Given that Y = n, X follows a binomial distribution
with parame
We now suppose that a team wins the series if it wins 2 games. X follows a negative binomial distribution with parameter 2 and 0.6. The probability mass function
of X is given by
i 1
P( X = i) =
0.62
Theoretical exercises
Exercise 1 : We verify that f is a density function. We perform an integration by
parts and get
Z
f ( x ) dx =
=
Let I =
I2
R +
0
=
=
e
Z
0
bx2
0
h
bx2
ax2 e
a
xe
2b
dx. We have
MATH 130A (Fall 2016)
Worksheet 2016.M.1a.s
Worksheet Solutions
Problem 1. Given a finite nonempty set X, prove that the number of subsets of X
with an even number of elements is equal to the number o
Fall 2015 Math 130A
Review problems
Exercise 1 :
A student owns 5 pairs of pants, 6 tee-shirts and 8 pairs of shoes. Every morning
they choose their clothes randomly and put on pants, a tee-shirt and
Fall 2015 Math 130A
Review problems
Exercise 1 : A pharmaceutical company decides to cut its advertising budget by only
sending three fifth of its mail as priority mail and the other two fifth as stan
Fall 2015 Math 130A
Review problems
Exercise 1 : Let X be an uniform random variable over [ a, b] with 0 < a < b. Let
Y = X2.
(a) Compute E(Y ) and Var(Y ).
Answer :
b3 a3 4( b a )4
,
.
3( b a )
45
(b
Spring 2015 Math 121A
- Review Problems Let F be a field.
Problem 1 : Are the following sets vector spaces?
(a) E1 = cfw_( a, b, c) R3 | a b = 2
Solution : No.
(b) E2 = cfw_ f F (R, R)| f (0) = 3
Solu
Exercise 13 :
A plane has three engines : one central and one on each wing. Each engine can
malfunction independently of the others. The central engine has a probability p to
malfunction and each wing
Fall 2015 Math 130A
Review problems
Exercise 1 : A pharmaceutical company decides to cut its advertising budget by only
sending three fifth of its mail as priority mail and the other two fifth as stan
Answer : E(Yn ) = 1 +
1
n
(0.85)n , n ln(0.85) >
ln n.
Exercise 4 : In a company, 60% of the objects are produced by a machine M1 and
the others by a machine M2. An object produced by M1 (resp. M2) is
Exercise 6 :
An urn contains n balls with the numbers 1 to n. We pick two balls without
replacement. Let X be the largest number of the two balls that we pick. For all k n
give P( X k) and P( X = k ).
(a) We pick without replacement six balls out of the urn. Let R and G be respectively the number of red balls and green balls that we pick. What distributions do R
and V follow? Give their expected va
(b) Find the probability density function of X.
8
> 0
>
for x 0
<
Answer : f X ( x ) =
.
1
> p e ln2 x/2 for x > 0
>
:
x 2p
(c) Compute E( X ).
p
Answer : e.
Exercise 4 : Let f : R ! R dened by
f (x)
Answer : n
n
1
.
k
k =1
Exercise 20 :
Cows have a certain disease D with probability p = 0.15. A farmer owns n cows.
To nd out if they have the disease D, the farmer has two possibilities:
Either he
Fall 2015 Math 130A
Homework solutions
Due Date
Sections
Problems
Thu, Oct 29.
Chapter 4 - Random Variable
1 , 13 , 17, 18, 20 , 22, 25 , 27, 29
Theoretical : 2 , 3
Exercise 1 : The possible values of
Exercise 20 : Let X denote your winnings when you quit. X : W ! F has the
following possible values
F = cfw_ 3, 1, 1.
X > 0 corresponds to X = 1. There are two possibilities for that. Either you win t
X = 500 corresponds to the probability to sell 1 standard encyclopedia. Thus
P( X = 500) = 0.3 0.5 (1
0.6) + (1
0.3) 0.6 0.5 = 0.27.
X = 1000 corresponds to the probability to sell 2 standard encyclop
(b) We rst check the cause 2, and then we check the cause 1. The possible values
for X are
cfw_ R2 + C2 , C2 + C1 + R1 .
X = C2 + R2 corresponds to the event that the second cause is the one responsib
Thus
E ( X ) = 3( p3 + (1
= 3(2p4
p)3 ) + 12( p(1
p )3 + (1
p) p3 ) + 30( p2 (1
p )3 + (1
p )2 p3 )
4p3 + p2 + p + 1).
Exercise 25 : Let X be the total number of heads. (a) X = 0 corresponds to both
c
Fall 2015 Math 130A
Homework solutions
Due Date
Sections
Problems
Thu, Nov. 5
Chapter 4 - Random Variable
33, 35 , 38, 39, 42, 44 , 49 , 57 , 60
Theoretical : 6, 10 , 15, 17
Exercise 33 : Let N be the