Solution to STAT200A Assignment 1
1. Let A and B be events for which P (A) and P (B) are both positive and less than one.
(a) Prove that if A and B are independent, then A B 6= .
(b) Prove that if A B = , then A and B are not independent.
Solution: (a) Su
STAT200A Assignment 1
1. Let A and B be events for which P (A) and P (B) are both positive and less than one.
(a) Prove that if A and B are independent, then A B 6= .
(b) Prove that if A B = , then A and B are not independent.
2. Let PP
on cfw_, F and ai
STATS 120B, W. Shen, Winter 2016
Hypothesis testing: what is it about?
Statistical inference: use data (sample) to study/describe the
population characteristics
Estimation: estimate the value of unknown parameters.
Point estimate, interval estimate
Wh
STATS 120B, W. Shen, Winter 2016
Topics for today
A few examples on calculating mgf
Revisit: covariance/correlation
Large-sample theory (Ch 6)
1
STATS 120B, W. Shen, Winter 2016
Kernel
I mean the function after removing constant from a pdf/pmf.
Examp
Stats 120B Homework 4: Due Thursday, Feb. 11
1. Let X and Y be two i.i.d random variables following N (3, 1).
(a) Find the distribution of 2X + Y using any method you like.
(b) Find the distribution of X 2Y using mgf. (Hint: the mgf of N (, 2 ) is
exp(t +
Stats 120B Homework 2: Due Thursday, Jan. 21
1. [10 pts] Let X be a random variable following Beta distribution Beta(a, b). Let Y
be another random variable following Poisson distribution with parameter . Find
the probability P(Y > X).
2. Let X1 and X2 be
Stats 120B HW 2: Solutions
1. [10 pts] Let X be a random variable following Beta distribution Beta(a, b). Let Y
be another random variable following Poisson distribution with parameter . Find
the probability P(Y > X).
Looking at the support of X and Y- X
Stats 120B/Math 131B: Midterm Exam, Winter 2016
Version A
Name:
Homework ID#:
UCI Student ID#:
Please read the following instructions carefully:
1. You must show your work/reasoning to receive full credit.
2. The exam is closed book and closed notes excep
Stats 120B Homework 7 Solution
1. Let X and Y follow a bivariate normal distribution with means (2, 1), variances
(1, 4) and covariance . In other words,
!
"
!
" !
"
X
2
1
N
,
Y
1
4
(a) Whats the marginal distribution of Y ?
N (1, 4)
(b) Whats P(X 2 = Y
Stats 120B Homework 1: Due Thursday, Jan. 14
1. Suppose that X has the gamma distribution with parameters and .
(a) [10 pts] Determine the mode of X. (Be careful about the range of )
To find the mode, take the log and then use the first derivative test (s
STATS 120B, W. Shen, Winter 2016
Several common mistakes from the midterm
(1) WLLN, CLT, use
p
d
Should be using and
(2) Zero covariance implies independence
Need to say bivariate normal distribution
(3) Likelihood. Write
Should write n
Q
Qn
i=1 f
STATS 120B, W. Shen, Winter 2016
HW4 Problem 3
Suppose that X1 , . . . , Xn form a random sample from a
uniform distribution with the following pdf
f (x|) =
1
I( x 2)
(f) Whats the mode of f (x|)?
Find x that maximizes f .
Any x between and 2 will do.
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STAT200A Assignment 3
1 Let X~_N(,u1,al), Y~IN(1L2,0), X and Yare independent. Dene U: X-lY, V: X. Y. Find
_-tl1e necessei y and sufcient condition for U and V to be independent.
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STAT200A Assignment 2
1. Suppose that 5% men and 0.25% women are color-blinded. A person is chosen at random and that
person is color-blinded. What is the probability that the person is male? (Assume males and females
to be equal in numbers.)
Solution: De
STAT200A Midterm
Some distribution functions relevant to the exam:
Let X P oisson() where > 0, then P (X = x) =
x e
x! , x
= 0, 1, .
Let X Exponential() where > 0, then its pdf is f (x) = ex , x > 0.
Let X N (, 2 ) where 2 > 0, then its moment generati
STAT200A Assignment 3
1. Let X N (1 , 12 ), Y N (2 , 22 ), X and Y are independent. Define U = X + Y , V = X Y . Find
the necessary and sufficient condition for U and V to be independent.
p
2. let X and Y be independent random variables with X N (0, 1) an
STAT200A Assignment 2
1. Suppose that 5% men and 0.25% women are color-blinded. A person is chosen at random and that
person is color-blinded. What is the probability that the person is male? (Assume males and females
to be equal in numbers.)
2. Let X be
STAT200A Assignment 4 Due on Friday, Dec 4, 2015
1. Let X N3 (0, ) with
1
=
1
, where 1/2 < < 1.
1
(a) Give the joint distribution of the first two variables given the third.
(b) What is the correlation of the first two variables given the third? What
Stats 120B Homework 1: Due Wed, Jan. 18
1. Let X and Y be two independent random variables following beta distributions
Beta(120, 2017).
(a) [5 pts] Whats P(X = 0.5)?
(b) [5 pts] Whats P(X + 3 < 2Y )?
(c) [5 pts] Whats P(X > Y )?
2. Suppose that X has the
Stats 120B: Jan, 09&11,2017
Prof: Weining Shen (weinings, 2204 Bren Hall)
TA: Fan Yin (yinf2)
Reader: Wei Hu (huw5)
About this class
All the materials are on the class website.
One midterm, one final
HW: once a week, due on Wed. (HW number)
Office hour: M
Stats 120B: Jan, 09&11,2017
Prof: Weining Shen (weinings, 2204 Bren Hall)
TA: Fan Yin (yinf2)
Reader: Wei Hu (huw5)
About this class
All the materials are on the class website.
One midterm, one final
HW: once a week, due on Wed. (HW number)
Office hour: M
> n
[1] 100
> mean(x)
[1] 1.49067
> vaer)
[1] 15.95129
> stat = (mean(x) - 0)/(sqrt(var(x)/(n)
> stat
[1] 3.732360
> pt(stat,n-1,lower=F) *2
[1] 0.0003167425
(f) Find the value of 71X, 82 and #0.
(g) State the alternative hypothesis.
(h) State your conclu