Problems
279
maximizing f (x1 , . . . , xn |); that is, it is just the maximum likelihood estimate of [when
it is constrained to be in (a, b)]. In other words, the maximum likelihood estimate equals
the mode of the posterior distribution when a uniform pr
Q&A
Based on the discussion after Monday class on May 6, 2013.
If we have independent random variables X and Y then Cov(X, Y ) = 0. Is the
converse true? i.e. Can we say Cov(X, Y ) = 0 implies independence?
Answer is no. Here is a simple example, which t
MATH343-02 Spring 2013
1
(17, May, 2013)
Getting started with R
1. (Install R) If you already installed R you may skip this step. Visit http:/cran.r-project.
org/bin/windows/base/ and click Download R 3.0.0 for Windows.
2. (Launch R) Once the installation
6 . .
120 numbers are rounded o to the nearest integer and then summed. If the individual roundo
errors are uniformly distributed between 0.5 and +0.5, what is the approximate probability
that the resultant sum diers from the exact sum by more than 3?
Answer key may contain errors. Use it at your own risk. Also,
remember these are only partial answers.
Prob 7. (a) |50 n X 8.2| > 1.96.
(b) n=6.
(c) reject.
(d) Approximately 1.
Prob 12. (a) pvalue suggest that the new toothpaste results in fewer than 3
c
Chapter 4 Homework Hint and Partial Answer
Prob 6. You need to nd the value of rst. Also note f (x) = 0 when x < 0.
P (50 X 150) 0.38 and P (0 X 100) 0.63.
Prob 12. Note there is a typo in the problem.
(a) fX (x) = xex .
(b) fY (y) = ey .
(c) You can easi
Chapter 5 Homework Hint and Partial Answer
Prob 6. X is Binomial (10,0.7) distributed.
10
0.74 0.36 .
4
(a) P cfw_X = 4 =
(b) P cfw_X > 12 = 0.
i
Prob 7. (a) P cfw_X i =
k=0
n k
p (1 p)nk .
k
n
P cfw_Y n i =
k =ni
n
k =ni
n
(1 p)k pnk . Let k = n k . Then
Answer key may contain errors. Use it at your own risk. Also,
remember these are only partial answers.
Prob 1. The maximum likelihood estimator is, min (xi ).
Prob 2. =
2n
n
i=1
xi
.
Prob 3. Maximum Likelihood Estimator of 2 is
(xi )2 /n. Its mean is 2 .
336
Chapter 8: Hypothesis Testing
Problems
1. Consider a trial in which a jury must decide between the hypothesis that the
defendant is guilty and the hypothesis that he or she is innocent.
(a) In the framework of hypothesis testing and the U.S. legal sys
Problems
223
SOLUTION If we let Xi be the amount consumed by the ith member of the sample,
i = 1, . . . , 25, then the desired probability is
P
X1 + + X25
> 150 = Pcfw_X > 150
25
where X is the sample mean of the 25 sample values. Since we can regard the
GENERATING AND GRAPHING A BINOMIAL PROBABILITY
DISTRIBUTION USING EXCEL
Suppose you wish to generate the probability distribution for a binomial random variable
with parameter (N=5, p=0.3). Do the following:
1.Input the following numbers in the spreadshee