Math 215 Homework (do by 12/1, turn in 12/2)
Turn in:
Problem 1 If
X=
a
b
Y =
c
d
and
then nd X T Y .
Problem 2 If
a
b
Y =
c
d
then write a2 + b2 + c2 + d2 using matrix notation with Y .
Problem 3 You are given the points (1, 8), (2, 5), (3, 3), (4, 4
Math 215 Midterm 3 Answers
Problem 1 (5 pts) For the following data, find the median: 9, 19, 2, 22, 7, 6 If we write these in order we get 2, 6, 7, 9, 19, 22 and the middle two numbers are 7 and 9. So the median is
7+9 2
= 8.
Problem 2 (5 pts) If P r(E) =
Math 215 Practice Final Exam
Problem 1 (10 pts) Find all rst-order partial derivatives of the following functions: f (x, y) = 7x10 y + 2x + y 2
fx fy
= 70x9 y + 2 = 7x10 + 2y
g(x, y, z) = xexy + yz ln x
gx gy gz
= = =
exy + xyexy + x2 exy + z ln x y ln x
Math 215 Practice Final Exam
Problem 1 (10 pts) Find all first-order partial derivatives of the following functions: f (x, y) = 7x10 y + 2x + y 2 g(x, y, z) = xexy + yz ln x
Problem 2 (10 pts) True or False: T T T T T F F The probability of an event is ne
Math 215 Final Review 1 Topics, grouped according to similarity
Multivariable calculus Three dimensional pictures Reading three-dimensional graphs Contour graphs/Level curves Partial derivatives Interpreting the meaning of partial derivatives Calculating
SPME430
SAMPLE EXAM (II)
1.
A baseball player throws a baseball vertically upward in a gymnasium so that the baseball just reaches the
ceiling. The height of the ceiling in the gymnasium is 8 m above the level that he releases the baseball
(i.e, the verti
Math 510 midterm 3 answers
Problem 1 (10 pts) Suppose X and Y are independent exponential random
variables both with parameter = 1. Find the probability that Y < 7X.
7x
P (Y < 7X)
7x
f (x, y) dy dx
=
ex ey dy dx
=
0
0
7x
0
ex ey
=
0
dx
ex e7x + ex dx
=
0
Math 215 Practice Final Exam
Problem 1 (10 pts) Find all rst-order partial derivatives of the following
functions:
f (x, y) = 7x10 y + 2x + y 2
fx
fy
= 70x9 y + 2
= 7x10 + 2y
g(x, y, z) = xexy + yz ln x
gx
=
exy + xyexy +
gy
gz
=
=
yz
x
x2 exy + z ln x
y
Math 215 Practice Final Exam
Problem 1 (10 pts) Find all rst-order partial derivatives of the following
functions:
f (x, y) = 7x10 y + 2x + y 2
g(x, y, z) = xexy + yz ln x
Problem 2 (10 pts) True or False:
T
F
The probability of an event is never greater
Math 510 Final Review Sheet
1
Topics
1. Combinatorics
(a) Basic principle of counting: multiplication
(b) repeated choices from a set: with or without order, with or without
repetition
with order, with replacement: nr
with order, without replacement: perm
Math 510 HW #20 Selected Answers
Problem 1 Let X1 , . . . , Xn be independent, identically distributed (i.i.d.) random variables, each with expected value 3 and standard deviation 7.
Find the expected value and standard deviation of X = X1 +Xn .
n
E[X] =
Math 510 HW #22 Answers
Problem 1 Ch. 9 Theo. ex. p. 484 #1a
P (N (20) = 2 | N (60) = 2)
P (N (20) = 2 N (60) = 2)
P (N (60) = 2)
P (N (20) = 2) P (N (60) N (20) = 0)
P (N (60) = 2)
=
=
(20)2 20 1 40
0! e
2! e
(60)2 60
2! e
2
400 60
2! e
36002 60
2! e
=
Math 510 HW #21 Selected Answers
Problem 1 Ch. 8 Problems p. 457 #3
We assume X = X1 +Xn is normal, due to the Central Limit Theorem.
n
5
We use = E[X] = 75 and = SD(X) = n . We want
P (70 < X < 80) .9
If we balance the probability .9 around the mean, the
Math 510 HW #23 Answers
Problem 1 Ch. 9 p. 484, problem #4
There are 4 states, dened by the number of white balls in the rst urn.
The transition matrix is
0 1 0 0
1 4 1 0
9
9 9 4 1
0 4
9
9
9
0 0 1 0
Ch. 9 p. 484, problem #5
The transition matrix is
.
Math 510 HW #19 selected answers
Problem 1 Let X be a random variable with the following mass distribution:
0
1
2
.3
.5
.2
Find the moment generating function for X.
= E[etX ]
= .3 e0t + .5 e1t + .2 e2t
= .3 + .5 et + .2 e2t
MX (t)
Problem 2 Let X be a ra
Math 215 Homework 42, do by 11/25/08, turn in
12/1/08
Read Calc. 7.5 for 11/25
Problem 1 If
X=
2
1
3
0
3
7
then nd X T (X transpose).
Problem 2 For the matrix
1
X = 2
8
nd X T X.
Problem 3 Solve the following equation for X:
1
0
1
1
X=
3
9
Problem 4 Solv
Math 215 Midterm 2 Answers
Problem 1 (5 pts) Draw a Venn diagram illustrating
A (B C)
U
A
B
C
Problem 2 (5 pts) If we dene the following sets:
U
S
=
=
cfw_years from 1850 to the present
cfw_years that contained a recession
T
=
cfw_U.S. election years
Then