Homework 2
Due - 06/26/2017
Each question worth 5 points
1. Consider a gambling game of tossing a coin at a casino, where probability of
getting a head is 0.55. You and the casino both has infinite amount of money
to spend. Lets consider every time you do
STAT 416 Fall 2014
Homework 5 Solutions
November 3, 2014
4.20 Let R1 , R2 , and R3 denote the events of winning on the rst, second, and third spin respectively.
c
c
(a) P (X > 0) = P (R1 R1 R2 R3 ) = P (R1 ) + P (R1 ) P (R2 ) P (R3 ) =
0.5918.
18
38
+
20
Homework 4 Solutions
Alex Misiats
[email protected]
August 28, 2014
3.42 Let F denote the event the cake failed to rise. Then
P (A) P (F |A)
0.5 0.02
10
P (A|F ) =
=
=
.
P (A) P (F |A) + P (B) P (F |B) + P (C) P (F |C)
0.5 0.02 + 0.3 0.03 + 0.2 0.05
29
Homework 2 Solutions
Alex Misiats
[email protected]
August 28, 2014
1.24 Using the binomial theorem,
5
3x2 + y
5
=
i=0
5
i
3x2
i
y 5i = y 5 + 15x2 y 4 + 90x4 y 3 + 270x6 y 2 + 405x8 y + 243x10 .
1.28 There are four choices of school for each teacher. If
Homework 3 Solutions
Alex Misiats
[email protected]
August 28, 2014
3.6 Notice that in both cases, the probabilities of all four sequences of balls that contain exactly three
white balls, wwwb, wwbw, wbww, and bwww, are equally likely. Since only two of
STAT 416 Fall 2014
Homework 6 Solutions
November 3, 2014
4.30 Let Y denote the index of the rst tail ip. Then P (Y = n) = 2n for n = 1, 2, . . . , and 0 otherwise.
Note that X = 2Y . Thus
E [X] = E 2Y =
i=1
2i 2i =
P (Y = i) 2i =
i=1
1 = +.
i=1
(a) The pr
STAT 416 Fall 2014
Homework 7 Solutions
November 3, 2014
4.82 Let X be the number of nondefective transistors in the lot of 20. Since each transistor is defective with
probability 0.1, P (X = n) = 20 0.9n 0.120n .
n
20
P (lot rejected)
=
20
P (lot rejecte
STAT 416 Fall 2014
Homework 11 Solutions
December 9, 2014
6.26 (a) We are given that FA (x) = FB (x) = FC (x) = x for x (0, 1), 0 if x 0 Since A, B, and C are
mutually independent, so
FABC (x, y, z) = FA (a) FB (b) FC (c) = abc, (a, b, c) (0, 1) (0, 1) (0
STAT 416 Fall 2014
Homework 10 Solutions
December 3, 2014
6.6 The table for P (N1 = i, N2 = j):
HH
j
H
HH
i
1
2
3
4
1
2
3
4
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0
0.1
0.1
0
0
0.1
0
0
0
6.7 Since X1 is number of failures preceding the rst success, X1 + 1 is the ind
STAT 416 Fall 2014
Homework 9 Solutions
November 3, 2014
4.78 Each of the draws of 4 balls is a Bernoulli trial; we are interested in nding out whether the rst
success (drawing exactly 2 black balls out of 4) occurs on the n-th trial. If X is the index of
STAT 416 Fall 2014
Homework 8 Solutions
November 3, 2014
4.51 The number of typos on a page is approximately Poisson-distributed. Let X denote the number of
typos, then X P oisson (). Since = E [X] = 0.2, then
(a) P (X = 0) = e = e0.2 0.8187, and
(b) P (X
Homework 1
Due - 06/19/2017
Each question worth 5 points
1. Consider the game of tossing a fair coin
a. until 1 head occurs. Find the probability it takes 25 or less toss to stop
the game.
b. until 5 head occurs. Find the probability it takes 50 or less t
MATH 416 Fall 2014
Homework 1 Solution
August 28, 2014
1.1 (a) There are 26 choices of letters for each of the rst 2 places and 10 choices of digits for each of
the last 5 places. So by the basic principle of counting, there are 262 105 = 67600000 dierent