1
Sample Final Problems
1. (25 points) The number of daily trac accidents at a certain intersection is a random
variable that has a Poisson distribution. The mean of the Poisson distribution depends on
whether it is a week day (Mon.-Fri.) or weekend day (
131B HW#5 solution
7.8 Jointly Sucient Statistics
10. The p.d.f. of the uniform distribution is f (x|) = 1 1cfw_0x , so the likelihood is
fn (x|) =
n
f (xi |) =
i=1
1
1cfw_maxcfw_x1 ,.,xn 1cfw_0mincfw_x1 ,.,xn
n
It implies that the M.L.E. = maxcfw_X1 ,
Handout 6
STA 131B
Let X1 , . . . , Xn be a random sample from f (x|) and T = r(X1 , . . . , Xn ) be a statistic.
Definition of Sufficiency:
T is sufficient for if the conditional distribution of
(X1 , . . . , Xn ) given T = t does not involve (or is the
131B HW#1 solution
3.7 Multivariate Distributions
1. (a) We have
1
1
1
f (x1 , x2 , x3 )dx1 dx2 dx3 = 3c
0
0
0
Since the value of this integral must be equal to 1, it follows that c = 1/3.
(b) For 0 x1 1 and 0 x3 1,
1
1
f13 (x1 , x3 ) =
f (x1 , x2 , x3 )
STA 131B HW2
7.5
8. The likelihood function is
(
exp(n
0
fn (x|) =
Pn
i=1 xi )
for min(x1 , . . . , xn ) >
otherwise.
(a) For each value of x, fn (x|) will be a maximum when is made as large as possible subject
to the strict inequality < min(x1 , . . .
STA 131B
Handout 2
Winter 2016
Let cfw_Zn be a sequence of random variables.
Definition: Modes of Convergence
1. Zn Z in probability Zn Z 0 in probability.
For any > 0, P (|Zn Z| > ) 0, as n .
2. Zn Z in distribution Zn Z 0 in distribution.
FZn (t) FZ
Handout 3: Method of Moments
STA 131B
Definition of Method of Moments Estimator
This is another quick method for getting estimates that relies on a substitution idea. Let
X1 , . . . , Xn be a random sample from a population with parameter indexed by . Sup
Handout 1: Calculus Review
STA 131B
For more details, please see a calculus text.
1. Series
1an+1
if a 6= 1.
1a
P
1an+1
i
1 + a + a2 + =
i=0 a = limn 1a
P i1 P d i
d a
1
= i=1 da a = da
= (1a)
2,
i=1 ia
1a
a) 1 + a + + an =
b)
c)
P
i(i 1)ai2 =
P
ai
i=2
d
Do not start until instructed to do so. You can detach this page for scratch paper.
Midterm II
STA 131B
Winter 2016
University of California, Davis
Exam Rules: This exam is closed book and closed notes. You can use one page (two-sided)
of notes. Use of ca
STA 131A - Sample Final Solutions
Problem 1
a)
X is the number of accidents in the chosen day. P (weekday) = 5/7, P (weekend) = 2/7.
P (X > 1 | weekday)
=
=
P (X > 1 | weekend)
1 P (X = 0 | weekday) P (X = 1 | weekday)
0.06155
=
1 P (X = 0 | weekend) P (X
Final Review Topics:
1. Combinatorial Analysis
- Basic techniques for counting
- Equally likely outcomes
- Permutation
- Combination
2. Axioms of Probability
- Probability rules
3. Conditional Probability and Independence
- Conditional probability
- Rule
131B HW#6 solution
8.7 Unbiased Estimators
12. (a) Let X denote the value of the characteristic for a person chosen at random from
the total population, and let Ai denote the event that the person belongs to stratum i(i =
1, . . . , k). Then
= E(X) =
k
E