Probability for Engineering Applications
Mid Term Exam #1, October 20th, 2003
Last name of student
Student ID number
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Do This First:
1. Make sure that you have
12/5/03 3:25 PM
HW14 Solution page 1
Probability for Engineering Applications
SOLUTION Assignment #14
1. (1 pt) Express VAR[X+Y] in terms of E[X], E[Y], VAR[X], VAR[Y], and E[XY]. What happens if E
11/21/03 1:01 PM
HW 13page 1
Probability for Engineering Applications
SOLUTION: Assignment #13
Two random variables.
1. (2 pts) Let X be the input to a communications channel (cf 4.17). X takes on
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HW12, page 2
Joint empirical CDF of X and Y
1 0.8 0.6 0.4 0.2 0
19 ACTUAL 20 21 0 22 0.1 23 0.2 24 0.3 25 0.
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SOLUTION
Probability for Engineering Applications
Assignment #11
1. (3 pt) For the two-dimensional random variable Z= (X,Y) sketch the region of the plane corresponding to the following events and
10/31/03 9:01 AM
HW10 solution page 1
Probability for Engineering Applications
Solution to Assignment #10
1. (3 pts) The lifetime T (years) of a light bulb has pdf a. Find the reliability function
Solutions to Homework 08
1. Y = eX . We denote the pdf of X by fX (x) and its cdf by FX (x). Also, Y = eX X = ln Y . a. Note rst that for all values of X, i.e. < x < , Y > 0. Thus, P [Y 0] = 0. For
Solutions to Homework 07
1. a. In this case we have transmitted a 0 (i.e., v = 1). The receiver makes an error if the received signal is greater than 0 (i.e., if v + N 0). Then P [error|v = 1] = = =
Solutions to Homework 06
1. a. From the gure, fX (x) = Now, 1= =
a
0 c 1
|x| a
|x| > a |x| a
fX (x)dx
|x| dx a a a x c 1 =2 dx a 0 a x2 = 2c x 2a 0 = ac c 1 1 a b. FX (x) = 0 for x < a and FX
Solutions to Homework 05
1. 4 a. P [ace in rst draw] = 52 . b. Suppose the rst draw is seen to be an ace. Then, we now have 3 aces remaining in the 51 cards. Thus, 3 P [ace in second draw | ace in rst
Solutions to Homework 04
1. a. P [A B] = P [A] + P [B] P [A B] = P [A] + P [B] P [A]P [B]. b. P [A B] = P [A] + P [B] P [A B] = P [A] + P [B]. (since A B = P [A B] = 0) 2. a. This is a case
Solutions to Homework 03
[1] The number of ways of picking 20 raccoons out of N is N 20 The number of ways of picking 5 tagged raccoons out of 10 and 15 untagged raccoons out of N-10 is 10 N 10 5 15
Solutions to Homework 02
1. We are interested in the shaded region shown in Figure 1. The shaded are on the left corresponds to A B C while that on the right is AC B. Also, the unshaded area common
Solutions to Homework 01
1. One possible program: #include <stdio.h> #include <stdlib.h> #include <math.h> main(int argc, char *argv) { int i, alpha, old, new; if (argc != 3) { printf("Too few argumen
SOLUTION PEA Final Examination Thursday December 16, 2002
All six questions required for full grade. Please give explicit numerical answers wherever possible, on the same page as the question.
Q1 Q2
11/24/034:41 PM
Probability for Engineering Applications
Mid Term Exam #2, November 13th, 2003
Last name of student
Student ID number
First name of student
Email address
Do This First:
1. Make s
Probability for Engineering Applications
SOLUTION Assignment #15
The Central Limit Theorem has infinite applications. Here are four short examples. 1. (2 pts) A fair coin is tossed 1000 times. Estima