MATH 441 - MATHEMATICAL STATISTICS I
Mid Term II (Take Home Exam) Solutions
Problem 1: Find the constant K such that the following is a pdf
Kx2 , K < x < K
.
0,
otherwise
f (x) =
Solution: To be a valid pdf, we must have the following:
(1) fX (x) 0. That
SOLUTIONS
MATH 441 HOMEWORK 1
p.12[1,9(G)], p.18[7(U),9]
p.12 #1: Suppose that A B. Show that B c Ac .
Solution:
x B c x NOT B x NOT A because A B x Ac
.
QED
p.12 #9: [Graduate] Let S be a given sample space and let A1 , A2 , . . . be an innite sequence o
SOLUTIONS
MATH 441 HOMEWORK 5
p.164[2,5,8,9,10(U),12(G)]
p.164 #2: Suppose that a random variable X can have each of the seven values 3, 2, 1, 0, 1, 2, 3
with equal probability. Determine the pdf of Y = X 2 X.
Solution: For each value of x, we have the fo
SOLUTIONS
MATH 441 HOMEWORK 2
p.34[12,15,18], p.38[10], p.44[6(U),12(G)]
p.34 #12: Suppose that 35 people are divided in a random manner into two teams in such a way
that one team contain 10 people and the other team contains 25 people. What is the probab
SOLUTIONS
MATH 441 HOMEWORK 3
p.55[5], p.65[10,22], p.77[14,16]
p.55 #5: A box contains r red balls and b blue balls. One ball is selected at random and its
color is observed. The ball is then returned to the box and k additional balls of the same color a
SOLUTIONS
MATH 441 HOMEWORK 4
p.102[5], p.109[6,10], p.117[4(U),8(G),12]
p.102 #5: Suppose that a box contains seven red balls and three blue balls. If ve balls are selected at
random, without replacement, determine the pmf of X=the number of red balls th
MATH 441 HOMEWORK 6
Due by: Nov 5, 2008, 11 AM.
Fall 2008
1. The ow of trac at certain street corners can sometimes be modeled as a sequence of Bernoulli
trials by assuming that the probability of a car passing during any given second is a constant
p and
Math 441: Mid Term III (Take Home)
Fall 2008.
Name
Solve all problems. Show all your works to receive full credit.
1. (a) If X is normally distributed with mean 2 and variance 2, express P (|X 1| 2) in terms
of the standard normal cumulative distribution
Math 441: Mid Term II (Take Home)
Fall 2008.
Due by: October 17, 2008.
Name
noindent Solve all problems. Show all your works to receive full credit.
1. Find the constant K such that the following is a pdf
f (x) =
Kx2 K < x < K
.
0
Otherwise
1
2. The exper
Math 441: Mid Term III
Fall 2008.
Name: Solutions
Solve all problems. Show all your works to receive full credit.
1. Solve any two of the ve parts.
(a) If X is a random variable with Poisson distribution satisfying P (X = 0) = P (X = 1),
what is E(X)?
For
SOLUTIONS
MATH 441 HOMEWORK 6
#1: The ow of trac at certain street corners can sometimes be modeled as a sequence of
Bernoulli trials be assuming that the probability of a car passing during any given second is a
constant p and that there is no interactio
MATH 441 - MATHEMATICAL STATISTICS I
Mid Term III (Take Home) Solutions Due 11.17.8
Problem 1:
(a) If X is normally distributed with mean 2 and variance 2, express P (|X 1| 2) in terms
of the standard normal cumulative distribution function.
(b) If X is n
Math 335-01 October 27, 2014
PROBLEM SET 10
SOLUTIONS
1. Suppose the continuous random variable X has range [40,10] and probability density
function f with the property that ﬂ-x) =f(x) for all real numbers x. Let F be the
cumulative distribution function
Math 335-01 November 18, 2014
PROBLEM SET 13 SOLUTIONS
l. LetX and Y be independent random variables each having the probability density function:
x
_ _ 7
f(x)=l 2, OSxS.
0, otherwise
Calculate the probability density function of Z =X + l’.
71+: znueg 01:
Math 335-01 November 4, 2014
PROBLEM SET 11 SOLUTIONS
1. Suppose the continuous random variable X has an exponential distribution with expected value 0.4.
Let the discrete random variable Y be deﬁned as follows:
Y=k ifk-I EX‘Ekforle, 2. 3,
(3) Determine
Math 335-01 October 3, 2014
PROBLEM SET 6
Solutions
1. Suppose the discrete random variable X has the probability mass function:
0 1 otherwise.
(a) Determine the coEstant C. 9" 3 K _L :. g
r: ZCE) =CZ(?)= {-24. zC
Lao
END
_2.
=7C“?
I6 4 5' _ 5 HH_ 2 It]
(
Math 335-0 December 5, 2014
PROBLEM SET 14 SOLUTIONS
l. LetX and Y be jointly continuous random variables and have thejoint probability density function:
r) -2:
f(x’y)_{-e 4, 0<x<oo,0<y<x.
0, othenvise
For the random variable Z = 5X— 31’, compute (a) E(Z)
Math 335-01 September 3, 2014
PROBLEM SET 3
‘ I 50:. UT! 0 {U S .
Instructions: Complete solutions (make your reasonlng clear) for the followmg three
problems are due on September 10.
l. Prove that for every two events E and F, the probability that exactl
Math 335-01 October 21, 2014
PROBLEM SET 9
SOLUTIONS
1. Suppose that X I ,X 3 ,XP- - -,Xmare discrete random variables deﬁned on a countable
sample space+having the following propenies:
(i) E[X,]=3;i=l,2,3,-10
(ii) EleJ=68,i=l,2,3,-IO
_ 64 if |i—j|=l,i
Math 335-01 August 25, 2014
PROBLEM SET 1
.504. u 7‘ I 0 N5
Instructions: Complete solutions (make your reasoning clear) for the following two
problems are due on August 27 (but will be accepted without penalty on August 29).
I. In how many ways can 7 peo
Math 335-01 October 8, 2014
PROBLEM SET 7
SOLUTIONS
1. Suppose that X is a discrete random variable for which E[X] = 12 and Var[X] = 25. Compute
E[X(X-5)].
agar-5)] = 5;. Mac-5') For)
:. :éz—gx)pfr)
= Zx‘Pfx)‘ ‘5- Z‘rpx
K
= £152) # 5:213)
= V438?) +(£(Z)>
Math 335-01 October 14, 2014
PROBLEM SET 8 SOLUTIONS
1. Suppose thatX is a geometric random variable.
(a) Prove that for positive integers n and k, P(X =11 + k X > n): P(X = k).
w “w mm) = awe-JP mama—P2”
56 P(X:n+k}—Z,>n): Pﬁznrk’x>h)
I
f?
'U
to._/
5
Math 335-01 September 17, 2014
PROBLEM SET 4
SOLUTIONS
Instructions: Complete solutions (make your reasoning clear) for the following three
problems are due on Wednesday September 17.
1. Suppose on the upcoming weekend there is a 50% chance of rain on Sat
Math 335-01 September 24, 2014
PROBLEM SET 5
SOLUTIONS
1. Two students A and B are both registered for Math 335. Neither student has a sterling
attendance pattern. Student A attends class 75 percent of the time, and, even worse,
student B attends class on
Math 335-01 August 27, 2014
PROBLEM SET 2
59 4. u TAD/4.5
Instructions: Complete solutions (make your reasoning clear) for the following three
problems are due on September 3.
52
1. We know that there are [ 5 J possible ﬁve card poker hands.
(a) How many
Math 335-01 Nov. 14, 2014
PROBLEM SET 12 SOLUTIONS
l. The number of visits in a minute to the website for an online retail company has a Poisson distribution with
mean 20. A given visitor is a male aged 18 to 34 with probability 0.55, independent of other
Math 335-01
August 26, 2013
PROBLEM SET 1
Instructions: Complete solutions (make your reasoning clear) for the following two
problems are due on August 28 (but will be accepted without penalty on August 30).
1.
In how many ways can 9 people be seated in a