PSTAT 120B
Discussion 5 - Estimation
Spring 2016
1. Derivation of rows 3-4 of Table 8.1 through examples:
i.i.d.
i.i.d.
(a) Suppose X1 , ., Xn1 N (1 , 12 ) and Y1 , .Yn2 N (2 , 22 ) are independent Normal samples. Suggest an unbiased estimator for 1 2 and
PSTAT 120B
Discussion 4 - Sampling Distributions & CLT
Important Facts:
P
Sample Mean & Variance: Y = n1 ni=1 Yi ;
2
i.i.d.
Y1 , ., Yn N (, 2 ) Y N (, );
Spring 2016
Pn
2
i=1 (Yi Y )
(n1)S 2
Y
Y
N (0, 1);
tn1 ;
2n1
n
2
/ n
S/ n
Pn
i.i.d.
2
2
Y1 ,
PSTAT 120B
Discussion 1 - Review
Important Facts
CDF (page 158): FX (x) = P (X x);
X cont. fX (x) =
(k)
Spring 2016
d
dx FX (x)
MGF (page 139): MX (t) = E(etX ); MX (t)|t=0 = E(X k )
P
P (X = x) = all y P (X = x, Y = y)
R
Marginal PMF/PDF (page 236):
f
PSTAT 120B
Discussion 3 - Order Statistics
Spring 2016
Important Facts:
k
t
Poisson: Y Poisson() P (Y = k) = e k! ; MY (t) = e(e 1)
t
t1
1
1
; FY (t) = y
; MY (t) = et(22e
Uniform: Y Uniform(1 , 2 ) fY (y) = 2
1 )
1
2 1
Thm 6.5: Y1 , ., Yn i.i.d. wit
PSTAT 120B
Discussion 9 - Two Sample t-test
Spring 2016
1. Suppose we collect data on the length of time required to complete iPhone assembly using two
different manufacturing methods (data shown below). Is there sufficient evidence to indicate a
differen
PSTAT 120B
Discussion 7 - Pivotal Method
Spring 2016
Important Facts:
Pivotal Method:
(1) Find a pivotal quantity U that is function of Yi s and but has a probability distribution free of
. (The sufficient statistic for is often a good starting point).
(
PSTAT 120B
Discussion 1 - 120A Review and 120B Tips
Important Facts
CDF (page 158): FX (x) = P(X x);
MGF (page 139): MX (t) =
E(etX );
X cont. fX (x) =
(k)
MX (t)|t=0 =
P
Expected Value: (page 91, 170): E [g(X)] =
Indicator Functions:
1A =
all x
R
d
PSTAT 120B
Discussion 8 - Hypothesis Testing
Spring 2016
1. A tac is tossed is tossed 100 times and it lands on its side
times. Test the hypothesis that
the probability that the tac lands on its side is .1 against the alternative that it is greater than .